UC-NRLF 


75D 


SURVEYING  AND  TABLES 


BY 

G.  A.  WENTWORTH 

AUTHOR  OF  A  SERIES  6r  TEXT-BOOKS  IK  MATHEMATICS 


SECOND  REVISED  EDITION 


GINN  &  COMPANY 

BOSTON  •  NEW  YORK  .  CHICAGO  •  LONDON 


MATHEMATICAL   TEXT-BOOKS 

BY 

GEORGE  A.  WENTWORTH 


Elementary  Arithmetic 

Practical  Arithmetic 

Mental  Arithmetic 

Primary  Arithmetic    (Wentworth  and  Heed) 

Grammar  School  Arithmetic 

Advanced  Arithmetic 

Exercises  in  Arithmetic    (Wentworth  and  Hill) 

First  Steps  in  Algebra 

School  Algebra 

New  School  Algebra 

Higher  Algebra 

Elements  of  Algebra 

Complete  Algebra 

Shorter  Course  in  Algebra 

College  Algebra  (Revised  Edition) 

Exercises  in  Algebra    (Wentworth  and  Hill) 

First  Steps  in  Geometry    (Wentworth  and  Hill) 

Plane  and  Solid  Geometry  (Revised) 

Plane  Geometry  (Revised) 

Solid  Geometry  (Revised) 

Plane  and  Solid  Geometry  and  Plane  Trigonometry 

(Second  Revised  Edition) 
Analytic  Geometry 
Logarithms  and  Metric  Measures 
Geometrical  Exercises 
Syllabus  of  Geometry 

Examination  Manual  in  Geometry   (Wentworth  and  Hill) 
Exercise  Manual  in  Geometry    (Wentworth  and  Hill) 
Plane  Trigonometry  (Second  Revised  Edition) 
Plane   Trigonometry    and    Tables    (Second   Revised 

Edition) 
Plane  and  Spherical   Trigonometry  (Second  Revised 

Edition) 
Plane  and  Spherical  Trigonometry,  and  Tables 

(Second  Revised  Edition) 
Plane    Trigonometry    and    Surveying,    and    Tables 

(Second  Revised  Edition) 

Surveying  and  Tables  (Second  Revised  Edition) 
Plane  and  Spherical   Trigonometry  and  Surveying, 

and  Tables  (Second  Revised  Edition) 
Plane  and  Spherical  Trigonometry,  Surveying,  and 

Navigation  (Secpnd  Revised  Edition) 
Logarithmic  and  Trigonometric  Tables 

Seven  Tables  (Wentworth  and  Hill) 

Complete 


COPYRIGHT,  1882, 1895, 1896, 1903,  BY 
G.  A.  WENTWORTH 


ALL  BIGHTS  RESERVED 


PBEFACE 

THE  object  of  this  work  on  Surveying  is  to  present  the  subject  in 
a  clear  and  intelligible  way,  according  to  the  best  methods  in  actual 
use,  and  in  so  small  a  compass  that  students  in  general  will  find 
time  to  acquire  a  competent  knowledge  of  this  important  study. 

The  author  is  under  obligation  to  G.  A.  Hill,  A.M.,  of  Cambridge, 
Mass. ;  to  Professor  James  L.  Patterson,  of  Chestnut  Hill,  Pa. ;  to 
Dr.  F.  N.  Cole,  of  New  York,  N.Y. ;  to  Professor  S.  F.  Norris,  of 
Baltimore,  Md. ;  and  to  Professor  B.  F.  Yanney,  of  Alliance,  Ohio. 
Professor  Yanney  has  done  most  of  the  work  on  the  second  revision, 
and  Miss  M.  Gertrude  Cross,  of  Boston,  has  furnished  the  drawings. 

G.  A.  WEOTWORTH. 
EXETER,  N.H.,  1903. 


iii 


CONTENTS 

SURVEYING 

[The  numbers  refer  to  the  pages.] 

CHAPTER  I.     FIELD  INSTRUMENTS: 

Definitions,  1 ;  classification,  1 ;  operations  comprised,  2  ;  the  sur- 
veyor's chain,  3;  the  engineer's  chain,  3;  accompanying  pieces,  4; 
how  to  chain,  4 ;  special  constructions  by  means  of  the  chain,  5 ; 
obstacles  to  chaining,  7  ;  the  tape,  9 ;  the  compass,  10 ;  kinds  of 
compasses,  11 ;  bearing  of  a  line,  12  ;  checking  bearings,  13;  obsta- 
cles, 14  ;  measurement  of  horizontal  angles,  14 ;  measurement  of 
vertical  angles,  15;  verniers,  15;  uses  of  the  compass  vernier,  17; 
magnetic  declination,  19;  surveyor's  transit,  23;  uses,  24;  measure- 
ment of  horizontal  angles,  26 ;  measurement  of  vertical  angles,  26 ; 
stadia  measurements,  26  ;  the  solar  compass,  28 ;  to  establish  a  true 
meridian,  32  ;  the  Y  level,  36  ;  the  leveling  rod,  36  ;  substitutes  for 
the  Y  level,  39 ;  the  plane  table,  40 ;  to  orient  the  table,  42 ;  to  plot 
any  point,  43  ;  to  plot  a  field,  43  ;  the  three-point  problem,  44. 

CHAPTER  II.     OFFICE  INSTRUMENTS  : 

Definitions,  46  ;  the  diagonal  scale,  46 ;  the  circular  protractor,  47  ; 
constructions,  48  ;  the  planimeter,  49 ;  the  slide  rule,  49. 

CHAPTER  III.     LAND  SURVEYING: 

Definitions,  50 ;  special  methods  of  surveying,  and  of  computing 
areas,  51 ;  rectangular  system  of  co-ordinates,  52 ;  general  method 
for  farm  surveys,  57  ;  field  notes,  58 ;  computation  of  the  area,  58 ; 
balancing  the  work,  60 ;  supplying  omissions,  61  ;  to  make  a  plot, 
63  ;  modification  of  the  latitude  and  departure  method,  66 ;  location 
surveys,  67  ;  illustrative  problems,  67  ;  laying  out  the  public  lands, 
71 ;  reference  lines,  71 ;  townships,  71 ;  subdivision  of  townships, 
73  ;  meander  lines,  73. 


vi  CONTENTS 

CHAPTER  IV.     TRIANGULATION: 

Definitions,  74 ;  classification,  75 ;  measurement  of  base  lines,  75 ; 
measurement  of  angles,  76. 

CHAPTER  V.     LEVELING: 

Definitions,  77  ;  corrections  for  curvature  and  refraction,  77  ;  dif- 
ferential leveling,  78  ;  single  setting  of  the  level,  78  ;  several  settings 
of  the  level,  79 ;  profile  leveling,  80  ;  field  work,  81 ;  making  the 
profile,  84 ;  topographic  leveling,  85 ;  drainage  surveying,  86 ;  field 
work,  86  ;  plot  and  profile,  86. 

CHAPTER  VI.     RAILROAD  SURVEYING  : 

Laying  out  the  route,  89 ;  establishing  the  roadbed,  89 ;  excava- 
tions, 89  ;  embankments,  90  ;  curves,  91  ;  methods  of  laying  out  the 
curve,  92. 

CHAPTER  VII.     CITY  SURVEYING: 

Field-work  instruments,  94 ;  streets,  94  ;  blocks  and  lots,  96  ;  plots, 
96 ;  records,  96. 


SURVEYING 

CHAPTER   I 

FIELD  INSTRUMENTS 

SECTION  I 

DEFINITIONS 

Definition.  Surveying  is  the  art  of  determining  and  repre- 
senting distances,  areas,  and  the  relative  position  of  points  on 
the  surface  of  the  earth. 

Classification.  Of  surveying  there  are  various  kinds,  depend- 
ing upon  the  extent,  the  purpose,  or  the  method  of  the  survey. 
The  following  are  the  principal  divisions  : 

1.  Plane  Surveying,  in  which  the  part  of  the  earth's  surface 
surveyed  is  regarded  as  a  plane  ;  Geodetic  Surveying,  in  which 
the  true  figure  of  the  earth  is  regarded. 

2.  Land  Surveying,  in  which  boundary  lines,  contents,  and 
outline  maps  are  the  chief  things  aimed  at ;    Topographic  Sur- 
veying, in  which  differences  in  elevation  and  contour  maps  are 
chiefly  sought ;   Hydrographic  Surveying,  in  which  the  purpose 
is  to  determine  the  configuration  and  topography  of  the  bed  or 
basin  of  a  body  of  water  ;  Mine  Surveying,  in  which  the  posi- 
tion and  extent  of  underground  excavations  are  determined  and 
graphically  represented. 

3.  Rectangular  Surveying,    in  which  a  system  of  perpen- 
dicular lines  is  used  as  reference  lines  ;  Triangular  Surveying, 
which  proceeds  by  means  of  a  system  of  triangles  referred  to 
a  well  established  base  line. 

1 


2  SURVEYING 

Operations  Comprised.  Surveying  commonly  comprises  the 
following  three  distinct  operations : 

1.  The  Field  Measurements,  or  the  determining  certain  lines 
and  angles  by  direct  measurement. 

2.  The  Computation  of  the  required  parts  from  the  meas- 
ured lines  and  angles. 

3.  The  Plotting,  or  representing  on  paper  the  measured  and 
the  computed  parts  in  relative  extent  and  position. 

Historic  Note.  Surveying  is  undoubtedly  one  of  the  oldest  of  the  arts 
of  civilized  man.  The  Bible  contains  several  admonitions  not  to 
remove  "  the  ancient  landmark,"  as  in  Proverbs  xxii.  28.  To  the  Baby- 
lonians is  credited  the  division  of  the  circle  into  360  degrees.  The  Egyp- 
tians were  known  to  survey  frequently  the  valley  of  the  Nile,  a  necessity 
owing  to  the  periodic  overflow  of  that  river.  Thence  came  Geometry.  The 
Egyptians  also  possessed  rules  for  finding  the  area  of  land  of  various  shapes. 
Moreover,  on  Egyptian  soil  the  Greek  mathematician  Eratosthenes  made 
the  first  attempt  at  determining  the  circumference  of  the  earth  by  meas- 
uring an  arc  of  the  circumference.  This  was  in  276  B.C.  Among  the 
Romans  Surveying  was  considered  one  of  the  liberal  arts,  and  received 
impetus  in  the  time  of  Julius  Caesar  from  his  sweeping  order  that  the 
entire  empire  should  be  surveyed  for  the  purpose  of  equitable  adjustment 
of  taxes,  and  also  from  the  introduction  of  the  more  practical  parts  of 
Greek  Geometry.  The  works  of  Roman  surveyors  served  as  models  for 
centuries,  and  much  that  we  have  to-day  is  only  improvements  on  what  has 
been  handed  down  from  them.  For  a  brief  account  of  surveying  in  the 
United  States,  see  Cajori's  "The  Teaching  and  History  of  Mathematics 
in  the  United  States,"  pp.  92,  286. 


FIELD  INSTRUMENTS  3 

SECTION   II 
THE  CHAIN 

Surveyor's  Chain.  The  Surveyor's  Chain,  or  Gunter's  Chain 
as  it  is  often  called,  is  made  of  iron  or  steel  wire  and  is  4 
rods  or  66  feet  long,  composed  of  100  links  connected  by  small 
rings,  and  provided  with  a  tally  mark  at  the  end  of  every  10 
links.  A  link  as  a  unit  of  measure  includes  a  link  of  the  chain 
and  half  the  rings  that  connect  it  with  adjoining  links.  Each 
link  is  7.92  inches  long.  Since  a  chain  is  4  rods  long,  a  square 
chain  contains  16  square  rods,  and  since  an  acre  contains  160 
square  rods,  a  square  chain  is  one-tenth  of  an  acre.  A  square 
chain  contains  also  10,000  square  links  and,  therefore,  an  acre 
contains  100,000  square  links.  Hence,  if  a  given  area  is 
expressed  in  square  chains,  it  is  reduced  to  acres  by  pointing 
off  the  last  figure,  and,  if  expressed  in  square  links,  it  is  reduced 
to  acres  by  pointing  off  the  last  five  figures.  The  tally  marks 
are  appropriately  notched  to  facilitate  counting  links  from 
either  end,  the  one  at  the  middle  of  the  chain  being  rounded 
so  as  to  be  distinguished  readily  from  the  others.  Handles 
form  part  of  the  end  links,  to  which  they  are  so  attached  as 
to  prevent  twisting  and  to  allow  lengthening  or  shortening 
of  the  chain.  The  Surveyor's  Chain  is  used  in  measuring 
land. 

Engineer's  Chain.  The  Engineer's  Chain  differs  from  the 
ordinary  Surveyor's  Chain  chiefly  in  that  it  is  100  feet  in 
length,  the  length  of  each  link  being  1  foot.  It  is  used  in 
surveying  railroads  and  canals,  and  often  in  other  surveys 
where  extensive  lines  are  to  be  run. 

Both  the  Surveyor's  Chain  and  the  'Engineer's  Chain  are 
generally  provided  with  attachments,  so  that  from  the  full 
chains  half-chains  can  be  made  up,  to  be  used  in  case  of  rough 
or  hilly  country. 


4  SURVEYING 

Accompanying  Pieces.  Usually  eleven,  sometimes  ten,  Mark- 
ing pins  go  with  the  chain.  These  are  of  iron  or  steel,  about 
14  inches  long,  pointed  at  one  end  and  formed  into  a  ring  at 
the  other  end.  In  case  eleven  pins  are  used,  the  first  one  is 
placed  at  the  beginning  of  the  line  to  be  measured,  and  there- 
after one  at  the  end  of  each  chain.  The  last  pin  in  the  ground 
is,  therefore,  not  to  be  counted.  In  case  ten  pins  are  used, 
the  first  one  is  placed  at  the  end  of  the  first  chain,  and  so  on, 
the  last  pin  in  the  ground  being  counted.  Strips  of  red  cloth 
should  be  fastened  to  the  ring  ends  of  the  pins  so  as  to  make 
them  easily  visible.  Ranging  poles,  which  are  of  various 
lengths,  are  necessary  for  alignment.  These  are  commonly 
made  of  wood,  and  are  steel  shod,  graduated  to  feet,  and 
painted  in  alternate  red  and  white  stripes. 

How  to  chain.  Ranging  poles  should  be  placed,  one  at  each 
end  of  the  line  to  be  measured,  and  at  such  intermediate 
points  as  the  necessities  of  the  case  require.  A  head  chain- 
man  or  leader,  and  a  rear  chainman  or  follower  are  required. 
The  follower  takes  one  end  of  the  chain,  and  one  pin,  which 
he  thrusts  into  the  ground  at  the  beginning  of  the  line.  The 
leader  takes  the  other  end  of  the  chain  and  the  remaining  ten 
pins,  and  moves  forward  until  the  word  "  Halt "  from  the  fol- 
lower warns  him  that  he  has  advanced  nearly  the  length  of 
the  chain.  At  this  signal  he  stops,  and  the  follower,  mean- 
while having  placed  his  end  of  the  chain  against  the  pin  at 
the  beginning  of  the  line,  directs  the  leader  by  the  words 
"Right"  and  "Left"  until  he  is  exactly  in  the  line.  This 
being  accomplished,  and  the  chain  tightly  stretched  in  a  hori- 
zontal position,  the  follower  says,  "  Down."  The  leader  then 
puts  in  a  pin  at  the  end  of  the  chain  and  answers,  "  Down  " ; 
after  which  the  follower  withdraws  the  pin  at  his  end  of  the 
chain,  and  the  chainmen  move  forward,  repeating  the  process 
just  described  until  the  end  of  the  line  is  reached. 

If  the  marking  pins  in  the  hands  of  the  leader  are  all  placed 


FIELD   INSTRUMENTS  5 

before  the  end  of  the  line  is  reached,  after  putting  the  last  pin 
in  the  ground  he  waits  until  the  follower  comes  up  to  him, 
gives  him  the  ten  pins  in  his  hands  and  records  the  fact  that 
ten  chains  have  been  measured.  The  measuring  then  proceeds 
as  before.  If  the  distance  from  the  last  pin  to  the  end  of  the 
line  is  less  than  a  chain,  the  leader  places  his  end  of  the  chain 
at  the  end  of  the  line,  and  the  follower  stretches  tightly  such 
part  of  the  chain  as  is  necessary  to  reach  the  last  pin,  and 
the  number  of  links  is  counted.  If  the  ground  slopes,  one  end 
of  the. chain  must  be  raised  until  the  horizontal  position  is 
attained.  By  means  of  a  plumb  line  or  a  slender  staff  or,  less 
accurately,  in  case  of  the  leader  by  dropping  a  pin  (heavy  end 
downwards),  the  point  vertically  under  the  raised  end  of  the 
chain  may  be  determined!.  If  the  slope  is  considerable,  half 
a  chain  or  less  may  be  used  ;  in  which  case  care  must  be  taken 
that  the  correct  number  of  full  chains  and  links  is  found.  For 
instance,  if  a  tally  shows  15  half  chains  and  35  links,  the 
appropriate  measure  is  7  chains  and  85  links,  or,  as  it  is 
usually  expressed,  7.85  chains. 

Special  Constructions  by  Means  of  the  Chain.  1.  At  a  given 
point  in  a  given  line  to  construct  a  perpendicular  to  that 
line. 

Let  LE  (Fig.  1)  be  the  given  line,  and  P  the  given  point. 
On  LE  measure  off  PB  =  PA  =  20  links.  Then  place  one  end  of 
the  chain  at  B  and  the  other  end  at  A. 


Stretch  the  chain  from  the  middle  point,  /  \ 


and  mark  that  point,  as  C.     PC  is  the 
perpendicular  required.      (Why  ?)  / 


Or,  make  PB  =  30  links.     Place  one 


E 


A          P         B 
end  of  the  chain  at  P,  and  the  end  of 

the  90th  link  at  B.     Then,  taking  the 

chain  at  the  end  of  the  40th  link  from  P  and  stretching 
both  portions  tightly,  mark  that  point,  as  C.  Then  PC  is  the 
perpendicular  required.  (Why  ?) 


6  SURVEYING 

2.  Through  a  given  point  without  a  given  line  to  construct 
a  perpendicular  to  that  line. 

Let  LE  (Fig.  1)  be  the  given  line,  and  C  the  given  point. 
Take  any  point  as  B  in  the  line  and  stretch  the  chain  between 
C  and  B ;  then  swing  the  chain  about  C  until  the  point  at  B  is 
again  in  the  line,  as  at  A.  Measure  the  distance  between  A 
and  B.  Then  P,  the  mid-point  of  AB,  is  a  second  point  in  the 
required  perpendicular.  (Why  ?) 

Or,  let  the  middle  of  the  part  of  the  chain  between  C  and  B 
be  held  in  place,  and  swing  the  end  at  C  until  it  meets  the 
line  as  at  P.  PC  is  the  required  perpendicular.  (Why  ?) 

3.  At  a  given  point  in  a  given  line  to  construct  an  angle 
equal  to  a  given  angle. 


D      E 

FIG.  2 


Let  P  (Fig.  2)  be  the  given  point  in  the  given  line  LE, 
and  angle  A  the  given  angle.  Make  PD  =  AB.  At  D  and  B, 
respectively,  construct  perpendiculars  DF  and  EC.  Make 
DO  =  BC.  Then  angle  OPD  is  the  angle  required.  (Why  ?) 

4.  To  construct  any  given  angle,  as  25°  40'. 

Find  from  the  tables  the  tangent  of  25°  40',  which  is  0.4806. 
Lay  off  PD  (Fig.  .2)  =  100  links.  Construct  the  perpendicular 
DF  and  lay  off  DO  =  48.06  links.  Then  angle  OPD  is  the 
angle  required.  (Why  ?) 

5.  Through  a  given  point  to  construct  a  line  parallel  to  a 
given  line. 

Let  P  (Fig.  3)  represent  the  given  point,  and  LE  the 
given  line.  Through  P  lay  out  any  convenient  line  as  BA 


FIELD   INSTRUMENTS  7 

intersecting  LE.     Construct  angle  BPD  =  angle  PAE.     Then 
the  line  CD  is  the  required  line.     (Why  ?) 

/B 

C  P/  D 


E 


FIG.  3 

Obstacles  to  chaining.  In  general  practice  various  obstacles 
are  encountered  in  chaining.  The  circumstances  in  each  case 
must  decide  the  best  method  to  be  used.  Only  a  few  sugges- 
tive cases  can  be  considered  in  this  work. 

1.  To  measure  a  line  when  a  building,  or  other  object, 
stands  in  the  way. 

In  Fig.  4  construct  the 

perpendicular  AB,  the  per-    -^ 

pendicular  BC,  the  perpen- 
dicular CD  =  AB,  then  the 


D 


E 


perpendicular    DE,    which  FIG.  4 

will  be  in  line  LA  prolonged. 

Then,  LA  +BC  +  DE  =  LE.     (Why  ?)     As  a  check,  another 
series  of  perpendiculars  may  be  constructed. 
2.  To  measure  across  a  body  of  water. 

At  A   (Fig.  5)  lay 
out  AP,  making  angle 
P,4£=:600.     This  can 
be  done  by  laying  out 
the  equilateral  tri- 
angle   ABD.       At    P 
: range   out   PC,   mak- 
ing angle  APC  =  60°. 
FIG.  5  Then     measure     A  P. 


B 


8  SURVEYING 

The  line  AC  is  equal  to  AP.  (Why  ?)  If  C  is  some  fixed 
point  in  LE,  across  the  stream,  accessible  or  inaccessible,  we 
may  proceed  as  follows :  After  laying  out  AP,  as  already 
described,  with  90  links  of  the  chain  stretched  in  the  form 
of  an  equilateral  triangle,  and  with  one  side  of  this  triangle 
in  AP,  move  the  triangle  until  the  point  C  is  in  line  with 
the  forward  side  of  the  triangle.  Then  proceed  as  before. 

3.  To  measure  a  line  the  end  of  which  is  invisible  from 
the  beginning,  and  the  intermediate  points  are  unknown. 

C 

L D!  K 


FIG.  6 

Let  LE  (Fig.  6)  represent  the  line.  Lay  out  the  line  LR  so 
that  R  shall  be  beyond  E  and  visible  from  L.  Construct 
from  E  the  perpendicular  EA  to  LR.  Measure  LA  and  AE. 
LE  can  then  be  computed.  (How  ?)  If  intermediate  points 
on  LE  are  to  be  sought,  take  any  point  in  LA,  as  B ;  construct 
EC  perpendicular  to  LA  ;  then  measure  off  BD  of  such  length 
that  BD:AE  =  LB:LA.  The  line  LR  is  called  a  random 

L^__ E  line. 

4.    To    measure    the   dis- 
tance between  two  inaccessi- 
/  ble  points. 

N£v'  Let  L  and  E  (Fig.  7)  be 

/     \  two  inaccessible  points. 

/  \  Select  some  point  asP  from 

/  XN\N  which   both    L    and    E   are 

visible.  Measure  PL  and  PE 


E/  '   by  the  method  in  2.    Kange 


FIELD   INSTRUMENTS  9 

out  PL'  in  line  with  LP  and  equal  to  LP ;  similarly,  RE'  =  ER. 
Then  measure  L'E',  which  is  equal  to  LE.     (Why  ?) 

EXERCISE   I 

1.  Range  out  a  line  which,  by  estimation,  is  more  than 
10  chains  long.      Then  measure  it  with  the  chain  out  and 
back. 

2.  Prolong  a  line  beyond  a  building,  or  other  obstacle  which 
prevents  continuous  alignment. 

3.  Find  the  distance  from  a  point  to  a  line  when  the  dis- 
tance is  more  than  a  chain. 

4.  Lay  out  a  square  field  each  side  of  which  shall  be  5.76 
chains  long. 

5.  Find  the  length  of  a  line  by  means  of  a  random  line. 
Then,  as  a  check,  find  its  length  by  direct  measurement. 

SECTION  III 
THE   TAPE 

Kinds  of  Tape.  The  tape  measure  used  by  the  surveyor 
or  engineer  consists  of  a  thin  ribbon  of  steel,  or  of  linen 
with  interwoven  wires  of  brass,  wound  upon  a  reel,  often  in 
a  leather  or  metal  case.  Tapes  vary  in  length  from  25  feet 
to  500  feet  or  more.  They  are  variously  graduated  to  links, 
to  feet  and  inches,  to  feet  and  tenths  of  a  foot,  to  metric  units, 
or  to  a  combination  of  these.  A  common  combination  is  feet 
and  tenths  of  a  foot  on  one  side,  and  links  on  the  reverse  side. 

Uses.  The  kind  of  tape  determines  to  a  great  extent  the 
use  to  which  it  is  to  be  put.  If  33  feet  or  66  feet  long  and 
graduated  to  links,  the  evident  purpose  is  for  land  survey- 
ing. If  50  feet  or  100  feet  long  and  graduated  to  feet  and 


10 


SURVEYING 


tenths  of  a  foot,  the  tape  is  especially  designed  for  city  work 
Other  kinds  are  employed  in  bridge,  road,  or  mining  work,  in 
very  accurate  measurements  of  base  lines,  or  as  standards  of 
comparison  for  other  instruments  of  measurement. 


SECTION   IV 
THE   COMPASS 

Parts  and  their  Uses.  The  essentials  of  the  compass,  one 
style  of  which  is  shown  in  Fig.  8,  are :  the  compass  circle, 
graduated  to  half  degrees  and  figured  from  0°  to  90°  each  way 


FIG.  8.    THE  SURVEYOR'S  COMPASS 

NOTE.  The  letters  E  and  W  on  the  face  of  the  compass  are  reversed 
from  their  true  positions.  The  reason  for  this  is  that  if  the  sights  are 
turned  towards  the  west,  the  north  end  of  the  needle  is  turned  towards 
the  letter  W,  and  if  the  north  end  of  the  needle  is  turned  towards  E,  the 
sights  are  turned  towards  the  east. 

If  the  north  end  of  the  needle  points  exactly  towards  E  or  W,  the 
sights  range  east  or  west. 


FIELD   INSTRUMENTS  11 

from  the  north  and  south  points,  for  indicating  the  directions 
of  lines ;  the  magnetic  needle,  pivoted  on  a  pin  at  the  centre 
of  the  compass  circle,  for  showing  the  direction  of  the  mag- 
netic meridian ;  and  the  sight  standards,  attached  to  the  ends 
of  the  main  plate,  for  alignment.  To  the  main  plate  are 
attached  two  spirit  levels  at  right  angles  to  each  other  for 
leveling  the  instrument;  underneath  is  a  needle-lifting  screw 
which,  by  actuating  a  concealed  spring,  lifts  the  needle  from 
the  pivot  and  presses  it  against  the  glass  covering  of  the 
compass  circle  when  the  instrument  is  not  in  use ;  a  tangent 
screw,  and  almost  directly  under  it  a  clamp  screw,  which 
operates  the  vernier ;  and  a  small  dial  plate  for  keeping  tally 
in  chaining.  The  north  end  of  the  needle  usually  has  some 
ornamentation  to  distinguish  it  from  the  south  end,  and  a 
coil  of  tine  wire  is  wound  on  the  south  end  to  prevent  the 
needle  from  dipping.  The  sight  standards  have  fine  slits 
nearly  their  whole  length,  with  circular  openings  at  intervals 
to  facilitate  sighting  upon  an  object ;  on  the  edges  of  the  north 
standard  are  tangent  scales  for  reading  vertical  angles,  and 
on  the  outside  of  the  south  standard  are  two  eyepieces  at  the 
same  distance  from  the  main  plate  as  the  zeros  of  the  tangent 
scales,  respectively.  The  telescopic  sight  is  an  attachment  to 
the  south  standard,  now  often  used.  The  instrument  entire 
turns  horizontally  upon  the  upper  end  of  a  ball  spindle,  the 
lower  end  of  which  rests  in  a  spherical  socket  in  the  top  of 
a  Jacob's  staff,  or  a  tripod,  which  supports  the  instrument. 
The  socket  of  the  compass  which  fits  to  the  ball  spindle  is 
provided  with  a  clamp  screw  and  a  spring  catch.  From  the 
centre  of  the  plate  at  the  top  of  the  tripod  a  plummet  is 
suspended  by  which  the  centre  of  the  compass  can  be  placed 
directly  over  a  definite  point  on  the  ground. 

Kinds  of  Compasses.  The  compass  described  is  the  vernier 
compass,  or  surveyor's  compass,  and  is  the  one  in  general  use. 
If  there  is  no  vernier  attachment,  the  compass  is  called  a  plain 


12 


SURVEYING 


compass  and  is  used  in  running  new  lines  and  the  preparation 
of  maps.  A  railroad  compass  has  all  the  features  of  the  vernier 
compass,  and  has  also  a  vernier  plate  and  graduated  limb  for 
measuring  horizontal  angles. 

Hints  on  the  Use  and  Care  of  Instruments.  The  instruments 
described  in  this  work  are  adjusted  by  the  maker.  If  they 
should  require  readjustment,  full  directions  will  be  found 
in  the  manual  furnished  with  the  instruments.  Before  begin- 
ning to  use  any  instrument,  make  a  thorough  study  of  its 
various  parts  and  their  uses.  In  moving  or  adjusting  any 
part  always  know  what  you  are  doing  and  why  you  are  doing 
it.  When  an  instrument  is  not  in  use  keep  it  in  a  place  that 
is  free  from  moisture  and  dust. 

Bearing  of  a  Line.  The  magnetic  'meridian  of  a  place  is  the 
direction  which  a  bar  magnet  assumes  when  freely  supported 

in  a  horizontal  position. 
The  magnetic  bearing  of  a 
line  is  the  angle  it  makes 
with  the  magnetic  merid- 
ian. To  take  the  bearing 
of  a  line  proceed  as  fol- 
lows :  Place  the  compass 
so  that  the  Jacob's  staff, 
or  plummet  of  the  tripod, 
is  directly  over  one  end  of 
the  line,  and  level  by  press- 
ing with  the  hands  on  the 
main  plate  until  the  bub- 
bles are  brought  to  the 
centres  of  the  spirit  levels. 
Turn  the  south  end  of  the  instrument  toward  you,  and  sight 
at  the  ranging  pole  at  the  other  end  of  the  line.  Eead  the 
bearing  from  the  north  end  of  the  needle.  First,  write  N.  or 
S.  according  as  the  north  end  of  the  needle  is  nearer  N.  or  S. 


FIG.  9 


FIELD   INSTRUMENTS  13 

of  the  compass  circle ;  secondly,  write  the  number  of  degrees 
between  the  north  end  of  the  needle  and  the  nearest  zero  mark  ; 
thirdly,  write  E.  or  W.  according  as  the  north  end  of  the  needle 
is  nearer  E.  or  W.  of  the  compass  circle.  Thus,  in  Fig.  9  (&),  the 
bearing  is  K  45°  W. ;  (&),  K  60°  E. ;  (c),  S.  60°  W. ;  (d),  S.  45°  E. 

If  the  needle  coincides  with  the  KS.  or  E.W.  line,  the 
bearing  is  1ST.,  S.,  E.,  or  W.  according  as  the  north  end  of  the 
needle  is  over  K,  S.,  E.,  or  W.  As  the  compass  circle  is 
divided  into  half  degrees,  the  bearing  may  be  determined 
pretty  accurately  to  quarter  degrees. 

It  will  be  noticed  that  the  letters  E  and  W  on  the  face  of 
the  compass  are  reversed  from  their  true  positions.  These  are 
so  placed  in  order  that  when  the  sights  are  turned  towards 
the  west  the  north  end  of  the  needle  will  point  towards  the 
letter  W,  or  if  the  sights  are  turned  towards  the  east,  the 
north  end  of  the  needle  will  point  towards  the  letter  E.  It 
turns  out  that  if  the  south  sight  standard  is  always  turned 
towards  the  observer,  the  reading  at  the  north  end  of  the 
needle  will  indicate  the  true  bearing  of  the  line.  Should  the 
north  sight  standard  be  turned  towards  the  observer,  the  read- 
ing at  the  south  end  of  the  needle  would  then  be  taken. 

Checking  Bearings.  When  the  bearing  of  a  line  has  been 
taken,  the  instrument  should  be  removed  to  the  other  end  of 
the  line  and  the  reverse  bearing  taken.  The  number  of 
degrees  should  be  the  same,  but  the  letters  should  be  reversed. 
For  instance,  if  the  direct  bearing  is  K  35f  °  W.,  the  reverse 
bearing  should  be  S.  35f  °  E.  In  case  the  reverse  bearing  is 
not  what  it  ought  to  be,  there  is  some  mistake,  or  some  local 
disturbance,  or  both.  To  detect  errors  a  second  trial  at  the 
direct  bearing  should  be  taken.  To  detect  local  disturbances 
take  the  direct  and  reverse  bearings  of  other  lines  ranged  out 
from  the  beginning  of  the  line  whose  bearing  is  sought.  If 
they  all  show  the  same  difference  between  their  two  respective 
bearings,  the  evidence  of  some  local  disturbance,  as  iron, 


14  SURVEYING 

iron  ore,  etc.,  is  pretty  conclusive.  In  this  case  the  true  bear- 
ing of  the  line  can  be  obtained  by  making  the  necessary  cor- 
rection. In  all  cases,  precautions  should  be  taken  to  have  the 
chain,  pins,  and  other  instruments  that  would  affect  the  direc- 
tion of  the  needle  sufficiently  removed  from  the  compass. 

Obstacles.  When  a  fence  or  other  obstruction  interferes 
with  placing  the  instrument  over  the  line  the  instrument  may 
be  placed  at  one  side,  the  ranging  pole  being  correspondingly 
placed  at  the  other  end.  If  one  end  of  the  line  cannot  be  seen 
from  the  other  end,  run  a  random  line.  Then  (Fig.  6,  p.  8) 
tan  EL  A  =  AE  -=-  LA,  whence  the  angle  EL  A  can  be  found. 
This  angle  combined  with  the  bearing  of  the  random  line  will 
give  the  bearing  required.  Or  some  point  can  be  selected 
from  which  the  ends  of  the  line  are  visible.  The  distances  to 
the  ends  may  be  measured,  and  the  angle  between  the  two 
auxiliary  lines  may  also  be  measured.  Of  the  triangle  thus 
formed,  the  angle  at  the  beginning  of  the  given  line  may  be 
computed,  and,  when  properly  combined  with  the  bearing  of  the 
first  auxiliary  line,  will  give  the  required  bearing.  If  a  single 
triangle  is  not  sufficient,  a  series  of  triangles  may  be  employed 
until  the  end  of  the  line  is  reached. 

Measurement  of  Horizontal  Angles.  To  measure  a  horizontal 
angle  by  means  of  the  needle,  place  the  compass  over  the 
vertex  of  the  angle,  take  the  bearing  of  each  line  separately, 
and  combine  these  bearings  according  to  the  following  rules, 
as  suggested  by  Eig.  10 : 

1.  If  the  first  letters  of  the  bearings  are  alike,  and  also  the 
last  letters,  find  the  difference  of  the  bearings. 

2.  If  the  first  letters  are  alike,  and  the  last  letters  unlike, 
add  the  bearings. 

3.  If  the -first  letters  are  unlike,  and  the  last  also  unlike, 
subtract  the  difference  of  the  bearings  from  180°. 

4.  If  the  first  letters  are  unlike,  and  the  last  alike,  subtract 
the  sum  of  the  bearings  from  180°. 


FIELD   INSTRUMENTS 


15 


FIG.  10 


W 


Measurement  of  Vertical  Angles.  A  vertical  angle  is  an 
angle  the  sides  of  which  are  in  a  vertical  plane.  If  one  side 
of  a  vertical  angle  is  horizontal  and  the  other  ascends,  the 
angle  is  called  an  angle  of  elevation;  if  one  side  is  horizontal 
and  the  other  descends,  the  angle  is  called  an  angle  of  depres- 
sion. To  measure  an  angle  of  elevation  by  means  of  the  com- 
pass, sight  through  the  lower  eyepiece  to  a  point  that  is  as 
far  above  the  point  whose  elevation  is  sought  as  the  instru- 
ment is  above  the  point  from  which  the  elevation  is  to  be 
taken.  Kead  off  the  degrees  of  the  right-hand  tangent  scale, 
marked  by  a  card  placed  squarely  across  the  face  of  the  south 
standard,  the  top  of  the  card  being  in  the  line  of  sight.  To 
measure  an  angle  of  depression,  proceed  in  the  same  manner, 
using  the  upper  eyepiece  and  the  left-hand  tangent  scale.  If 
the  compass  is  provided  with  a  telescopic  sight  that  has  a 
vertical  circle  attachment,  these  should  be  used  instead  of 
the  eyepieces  and  tangent  scales. 

Verniers.  A  vernier  is  a  contrivance  for  measuring  portions 
smaller  than  those  into  which  a  line  is  divided.  We  shall 
describe  two  kinds. 

Let  AB  (Fig.  11)  be  a  portion  of  a  line  graduated  to  tenths 
and  hundredths  of  a  foot.  VR  is  the  vernier. 

In  (a),  nine  parts  of  the  line  are  divided  into  ten  equal  parts 
on  the  vernier.  Hence,  a  division  on  the  vernier  is  less  than 
a  division  on  the  line  by  the  difference  between  T^  of  a  foot 
and  TL  of  T§^  of  a  foot,  or  T^Vtf  of  a  foot.  Now,  if  the  vernier 


16 


SURVEYING 


is  moved  so  that  1  of  the  vernier  coincides  with  1  of  the  scale,  it 
has  moved  over  a  space  equal  to  yo1^  of  a  foot.  If  the  vernier 
is  moved  so  that  2  of  the  vernier  coincides  with  2  of  the  scale, 
it  has  moved  over  a  space  equal  to  T^2ff F  of  a  foot ;  and  so  on. 
In  (7>),  6  of  the  vernier  coincides  with  9  of  the  scale,  which 
indicates  that  the  zero  of  the  vernier  has  moved  past  3  of  the 
scale  a  space  equal  to  TQGO  cr  °f  a  foot.  The  reading,  then,  is 


uuurumu 


L 


I,  I.I  ,1 


(w/iaaaaiaaaaan? 


o/raaaarr    rm    u 


~* <to          op 


(aaaaaiaaaaj.n( 


FIG.  11 


0.536  foot.  This  form  of  the  vernier  is  known  as  the  direct 
form,  since  the  figuring  on  the  vernier  proceeds  in  the  same 
direction  as  that  on  the  scale. 

In  (c),  eleven  parts  of  the  line  are  divided  into  ten  equal  parts 
on  the  vernier.  Hence,  a  division  on  the  vernier  is  greater 
than  a  division  on  the  line  by  the  difference  between  T^  of 
jfa  of  a  foot  and  yi^  of  a  foot,  or  TTJ^^  of  a  foot.  Now,  if  the 
vernier  is  moved  so  that  1  of  the  vernier  coincides  with  10 


FIELD   INSTRUMENTS 


17 


of  the  scale,  i.e.t  the  end  of  the  6th  tenth,  the  vernier  has 
moved  over  a  space  equal  to  TIJ^  of  a  foot.  If  the  vernier  is 
so  moved  that  2  of  the  vernier  coincides  with  9  of  the  scale, 
the  vernier  has  moved  over  a  space  equal  to  TQS^O  °f  a  ^°°^  5 
and  so  on. 

In  (d),  6  of  the  vernier  coincides  with  7  of  the  scale,  which  in- 
dicates that  the  zero  of  the  vernier  has  moved  past  3  of  the  scale 
a  space  equal  to  To6o  o  °^  a  f°°t.  The  reading  here  is  0.636  foot. 

This  form  of  the  vernier  is  known  as  the  retrograde  form, 
since  the  figuring  on  the  vernier  proceeds  in  the  opposite  direc- 
tion from  that  on  the  scale.  In  either  form  the  following 
rule  for  using  and  reading  the  vernier  may  be  adopted : 

Move  the  vernier  until  its  zero  line,  or  index,  is  at  the  point 
to  which  the  required  measurement  is  to  be  taken;  read  the 
main  scale  to  the  nearest  division  below  the  index,  and  that 
number  of  the  division  line  of  the  vernier  which  stands  opposite 
a  line  of  the  main  scale. 


FIG.  12 


Compass  Vernier  and  its  Uses.  Let  LL1  (Fig.  12)  repre- 
sent the  limb  of  the  compass  graduated  to  half  degrees,  and 
W  the  vernier  divided  into  thirty  equal  spaces,  equal  to 
twenty-nine  spaces  of  the  limb.  Then,  one  space  of  the  vernier 
is  less  than  one  space  of  the  limb  by  l'(=30'  —  ^  of  29x30f), 
and  the  reading  may  be  obtained  to  single  minutes. 


18 


SURVEYING 


In  Fig.  12  the  index,  or  zero,  of  the  vernier  stands  between 
32°  and  32°  30',  and  the  line  of  the  vernier  marked  9  coincides 
with  a  line  of  the  limb.  Hence,  the  reading  is  32°  9'. 

When  the  index  moves  from  the  zero  line  of  the  limb  in  a 
direction  opposite  to  that  in  which  run  the  numbers  of  the 
limb,  the  number  of  minutes  obtained  as  above  must  be  sub- 
tracted from  30'  to  obtain  the  minutes  required.  (Why  ?)  If, 
however,  the  vernier  is  made  double,  that  is,  if  it  has  thirty 
spaces  on  each  side  of  the  zero  line,  it  is  always  read  directly. 
The  usual  form  of  the  double  vernier,  shown  in  Fig.  13,  has 


only  fifteen  spaces  on  each  side  of  the  zero  line.  When  the 
vernier  is  turned  to  the  right  less  than  15'  past  a  division  line 
of  the  limb,  read  the  lower  figures  on  the  left  of  the  zero  line 
at  any  coincidence ;  if  moved  more  than  15'  past  a  division 
line  of  the  limb,  read  the  upper  figures  on  the  right  of  the 
zero  line  at  any  coincidence  ;  and  vice  versa.  In  this  form  of 
the  double  vernier  it  will  be  observed  that  the  spaces  on  the 
vernier  are  larger  than  those  on  the  limb,  since  the  30  equal 
spaces  of  the  former  are  equal  to  31  half-degree  spaces  of  the 
latter. 


FIELD   INSTRUMENTS  19 

Tho  most  important  use  of  the  vernier  compass  is  in  setting 
off  the  variation  of  the  needle  explained  just  below.  If  the 
variation  of  the  needle  at  any  place  is  known,  by  means  of  the 
vernier  screw  the  compass  circle  may  be  turned  through  an  arc 
equal  to  the  variation.  If  the  observer  stands  at  the  south 
end  of  the  instrument,  the  vernier  is  turned  to  the  right  or  left 
according  as  the  variation  is  west  or  east.  The  compass  now 
gives  the  bearings  of  the  lines  with  the  true  meridian. 

In  order  to  retrace  the  lines  of  an  old  survey,  turn  the 
sights  in  the  direction  of  a  known  line  and  move  the  vernier 
until  the  needle  indicates  the  old  bearing.  If  no  line  is  defi- 
nitely known,  the  change  of  variation  from  the  time  of  the 
old  survey  will  give  the  arc  to  be  set  off. 

Magnetic  Declination.  The  magnetic  declination,  or  varia- 
tion of  the  needle,  at  any  place  is  the  angle  which  the  magnetic 
meridian  makes  with  the  true  meridian,  or  north  and  south 
line.  The  variation  is  east  or  west,  according  as  the  north  end 
of  the  needle  lies  east  or  west  of  the  true  meridian.  Western 
variation  is  indicated  by  the  sign  +,  and  eastern  by  the  sign  — . 
The  kinds  of  magnetic  decimation  are  put  under  three  heads : 

1.  Irregular   variations,  which    are    sudden  deflections    of 
the  needle  due  to  magnetic  storms  or  other  causes  not  well 
understood. 

2.  Solar-diurnal  variations,  which  in  northern  latitudes  reach 
their  farthest  point  east  about  8  o'clock  A.M.,  and  their  farthest 
point  west  about  2  o'clock  P.M.,  varying  from  5'  in  the  winter 
in  some  localities  to  20'  in  the  summer  in  other  localities. 

3.  Secular  variation,  which  is  a  change  in  the  same  direction 
for  a  period  of  years,  then  in  the  opposite  direction  for  about 
the  same  time. 

It  is  not  accurately  known  how  long  it  takes  a  complete 
secular  variation  to  run  its  course,  but  from  data  already 
obtained  it  seems  probable  that  the  period  of  time  covered  is 
not  less  than  two  and  a  half  or  three  centuries. 


20 


SURVEYING 


O 
Si    a 


»HOO     '  i-  O  rH  o  O 


I  + 


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oftM^rH^r-l        '   ^   ,!«   r-l   r-i   rH  Jin   ,_   ,_,   ^       *  ^  ^        ^^^CO 


. 

T—  i  Ol  ^f  O  -^ 


t—  O?  O  r-  1 


rH   <M   O   rH   OS  -^   OS   O   O  t^rHOOOl^J  OCO  Ci   O   O   (N 

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»c5dSCOod«>"3COt^Oijt^»O^^c6ocOrHC5(Nr-5OrHrH 


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M^^'^COO5odl^t^(NOOOt^OI>^^^OOrHC<ioC<ioOOOrHO 

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FIELD   INSTRUMENTS 


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22  SURVEYING 

The  agonic  line,  or  line  of  no  variation,  is  a  line  joining 
those  places  at  which  the  magnetic  meridian  coincides  with 
the  true  meridian.  At  the  beginning  of  the  present  century 
this  line  crossed  the  United  States  in  an  irregular  way  from 
Michigan  to  South  Carolina.  It  is  gradually  moving  west- 
ward, so  that  the  variation  is  increasing  at  places  east  of  this 
line,  and  decreasing  at  places  west  of  the  line.  East  of  this 
line  the  variation  is  westerly,  and  west  of  this  line  the  varia- 
tion is  easterly.  Lines  that  join  places  of  equal  magnetic 
declination  are  called  isogonic  lines. 

Table  of  Magnetic  Declination.  On  pp.  20,  21  will  be  found 
a  table  showing  the  variation  in  magnetic  declination  at 
different  places  in  the  United  States  and  contiguous  territory 
during  the  nineteenth  century  ;  also  the  annual  change  for  the 
epoch  of  1900. 

EXERCISE    H 

1.  Lay  out  a  field  of  five  sides  and  take  the  bearings  and 
measures  of  the  sides  in  order,  beginning  at  the  most  westerly 
point  and  going  about  the  field  clockwise. 

2.  From  the  bearings  obtained  in  Example  1  find  the  value 
of  each  of  the  interior  angles.     What  is  their  sum  ? 

3.  Lay  out  the  field  the  bearings  and  distances  of  whose 
sides  are  given  in  Example  1  of  Exercise  YI,  p.  64. 

4.  Eange  out  a  line  whose  bearing  is  X.  38°  30'  W.,  and  at 
some  point  in  this  line  range  out  another  line  making  a  right 
angle  with  it.     What  is  the  bearing  of  the  second  line  ? 

5.  Set  up  the  compass  at  a  spot  near  which  there  is  known 
to  be   some   local  disturbance,  as   iron   in  a  building,  or   an 
iron  fence,  and  find  the  variation  of  the  needle  due  to  such 
disturbance. 


FIELD   INSTRUMENTS  23 

SECTION   Y 
THE   TRANSIT 

Surveyor's  Transit.  The  transit  is  the  most  important 
instrument  used  in  surveying.  There  are  many  modifications 
of  it,  each  adapted  to  its  own  particular  use.  All,  however, 
have  about  the  same  essential  features.  The  one  described 
here,  and  shown  in  Fig.  14,  is  the  surveyor's  transit,  the  one 
of  most  general  use.  The  essential  parts  are  the  telescope 
with  its  axis  and  two  standards,  the  circular  plates  with  their 
attachments,  the  sockets  upon  which  the  plates  revolve,  the 
leveling  head,  and  the  tripod.  Within  the  telescope  are  two 
fine  cross  wires,  at  right  angles  to  each  other,  whose  intersection 
determines  the  optical  axis,  or  line  of  collimation,  of  the  tele- 
scope. Under  the  telescope,  and  attached  to  it,  is  a  spirit 
level  by  which  horizontal  lines  may  be  run,  or  the  difference 
of  level  between  two  stations  be  found.  The  axis  of  the  tele- 
scope carries  a  vertical  circle  which  measures  vertical  angles 
to  single  minutes  by  means  of  a  vernier.  The  vernier  plate, 
which  carries  the  telescope  and  also  the  compass  circle,  has 
two  verniers  diametrically  opposite  to  each  other,  and  it  moves 
entirely  around  the  graduated  limb  of  the  main  plate.  The 
sockets  are  compound ;  the  interior  spindle  attached  to  the 
vernier  plate  turning  in  the  exterior  socket,  when  an  angle  is 
taken  on  the  limb,  but  when  the  plates  are  clamped  together 
the  exterior  socket  itself,  and  with  it  the  whole  instrument, 
revolves  in  the  socket  of  the  leveling  head.  The  transit  is 
leveled  by  four  leveling  screws  which  pass  through  a  plate 
firmly  fastened  to  the  ball  spindle  and  rest  in  small  sockets, 
these  resting  in  turn  on  the  upper  side  of  the  tripod  plate. 
On  the  underside  of  this  lower  or  tripod  plate  is  an  arrange- 
ment called  a  shifting  centre,  which  enables  the  surveyor  to 
change  the  position  of  the  vertical  axis  horizontally  without 


24  SURVEYING 

moving  the  tripod ;  besides  this  there  is,  if  specially  ordered, 
a  device  called  a  quick-leveling  attachment  to  bring  the  transit 
quickly  to  an  approximately  level  position  by  the  pressure  of 
the  hands  after  which  the  leveling  screws  are  used. 

Uses.  The  transit  may  be  used  for  all  the  purposes  for 
which  the  compass  may  be  used,  but  with  much  greater 
precision.  The  principal  use,  however,  is  in  measuring  hori- 
zontal angles  by  means  of  the  graduated  limb  and  verniers. 
It  may  be  used,  furthermore,  in  obtaining  differences  of  level ; 
also,  provided  there  is  the  attachment  to  the  telescope  known 
as  the  stadia,  in  measuring  distances,  especially  over  broken 
ground.  A  still  further  use,  when  the  transit  is  supplied 
with  what  is  known  as  a  gradienter  attachment,  is  in  fixing 
grades  as  well  as  measuring  distances. 

Getting  the  Transit  Ready.  The  instrument  should  be  set 
up  so  as  to  be  firm,  the  tripod  legs  being  pressed  into  the 
ground  until  the  plates  are  as  nearly  level  as  can  conveniently 
be  done  by  this  means.  For  the  subsequent  leveling  turn  the 
instrument  until  the  spirit  levels  on  the  vernier  plate  are 
parallel  to  the  vertical  planes  passing  through  opposite  pairs 
of  the  leveling  screws.  Take  hold  of  opposite  screw  heads 
with  the  thumb  and  forefinger  of  each  hand,  and  turn  both 
thumbs  in  or  out  as  is  necessary  to  bring  the  bubble  to  its 
proper  place,  the  left  thumb  always  moving  in  the  direction 
that  the  bubble  is  to  move.  For  precise  work,  in  addition  to 
leveling  by  the  leveling  screws,  it  is  advisable  to  level  the 
plates  by  the  telescope  level,  as  this  is  much  more  sensitive 
than  the  levels  on  the  plate.  In  this  operation  the  position 
of  the  level  on  the  telescope  must  be  observed  over  both  sets 
of  leveling  screws,  one  half  the  correction  being  made  by  the 
axis  tangent  screw,  the  other  half  by  the  leveling  screws. 
Before  an  observation  is  made  with  the  telescope,  the  eye- 
piece should  be  focused  by  its  pinion  until  the  cross  wires 
appear  distinct ;  the  object  glass  is  then  focused  by  its  pinion 


FIELD   INSTBUMENTS 


25 


FIG.  14.    THE  SUUVEYOR'C  TRANSIT 


26  SURVEYING 

until  the  object  to  be  observed  appears  well  defined.  This 
latter  process  must  be  repeated  when  the  distance  to  the 
object  is  changed. 

Measurement  of  Horizontal  Angles.  Place  the  instrument 
directly  over  the  vertex  of  the  angle,  and  level.  Set  the 
limb  at  zero  by  the  tangent  screw  of  the  plates,  and  turn  the 
telescope  in  the  direction  of  one  of  the  sides  of  the  angle, 
directing  it  to  the  object  by  the  tangent  screw  of  the  leveling 
head.  Then  unclamp  the  main  plate  and  turn  the  telescope 
until  it  is  in  the  direction  of  the  other  side  of  the  angle, 
and  read  the  angle  by  the  verniers.  The  object  of  the  two 
verniers  on  the  vernier  plate  is  to  correct  any  mistakes  that 
might  arise  from  the  want  either  of  exact  coincidence  in  the 
centres  of  the  verniers  and  the  limb  or  of  exact  graduations 
on  the  limb.  The  correct  reading  may  be  obtained  by  adding 
to  the  reading  of  one  vernier  the  supplement  of  the  reading  of 
the  other,  and  taking  half  their  sum. 

Measurement  of  Vertical  Angles.  Direct  the  telescope  to  the 
object ;  clamp,  and  read  the  angle  indicated  on  the  vertical 
circle  by  the  vernier.  The  angle  read  will  be  an  angle  of 
elevation  or  depression  as  the  case  may  be,  the  horizontal  line 
being  the  line  of  collimation  of  the  telescope  when  in  a  hori- 
zontal position. 

Stadia  Measurements.  As  already  stated  on  page  24,  the 
stadia  is  an  attachment  to  the  telescope  used  in  measuring 
distances,  especially  over  rough  ground.  It  consists  essen- 
tially of  two  horizontal  wires  fastened  to  small  movable 
slides,  and  so  adjusted  as  to  include  a  given  space,  say 
one  foot  on  a  rod  100  feet  distant.  These  wires  will  then 
include  two  feet  on  a  rod  200  feet  away,  or  a  half-foot  at  a 
distance  of  50  feet,  and  so  on.  Usually  the  instrument  is  so 
adjusted  that  the  zero  of  the  indicated  distance  is  in  front 
of  the  centre  of  the  instrument;  hence,  the  true  distance 
is  the  indicated  distance  plus  the  distance  of  this  zero  from 


FIELD   INSTRUMENTS  27 

the  centre  of  the  instrument.  This  latter  distance  is  deter- 
mined for  each  instrument  by  the  maker,  and  noted  on  a  card 
placed  on  the  inside  of  the  instrument  box.  It  is  known  as 
the  constant  of  the  instrument.  The  readings  are  taken  on  a 
rod,  specially  designed  for  the  purpose,  known  as  the  stadia 
rod.  This  is  graduated  to  feet,  and  tenths  and  hundredths  of 
a  foot.  Any  ordinary  leveling  rod,  if  similarly  graduated,  will 
answer  the  same  purpose.  Obviously  in  taking  stadia  meas- 
urements the  rod  must  always  be  held  at  right  angles  to 
the  line  of  sight.  This  statement  has  special  reference  to 
measurements  taken  up  or  down  hill-slopes.  In  this  case,  if 
horizontal  distance  is  required,  the  measured  distance  must 
be  multiplied  by  the  cosine  of  the  angle  of  elevation  or 
depression.  (Why  ?) 


EXERCISE   HI 

1.  By  means  of  the  transit,  measure  the  interior  angles  of 
the  field  of  Example  1,  Exercise  II,  p.  22,  and  compare  with 
the  results  obtained  in  Example  2  of  the  same  exercise. 

2.  Lay  out  the  entire  angular  magnitude  about  some  point 
into  four  or  more  angles,  and  measure  each  one  of  them.     What 
should  the  sum  of  them  equal  ? 

3.  If  the  constant  of  a  transit  adjusted  to  one  foot  100  feet 
away  is  3.8  inches,  what  is  the  true  length  of  a  line  when  the 
indication  on  the  rod  is  2.35  feet  ? 

4.  Measure  a  line  by  the  stadia,  and  compare  with  measure- 
ments taken  by  the  chain  and  also  by  the  tape. 

5.  Compute  the  height  of  a  tall  object,  as  a  tree  or  steeple, 
by  first  measuring  its  distance  from  some  convenient  point  and 
measuring  the  angle  of  elevation  at  that  point. 

6.  Lay  out  a  square  field  containing  just  one  acre. 


28  SURVEYING 

SECTION   VI 
THE    SOLAR    COMPASS 

Description  and  Uses.  A  full  description  of  the  solar  com- 
pass, or  Burtfs  solar  compass,  as  it  is  often  called  from  its 
inventor,  with  its  principles,  adjustments  and  uses,  forms  the 
subject  of  a  considerable  volume,  which  should  be  in  the  hands 
of  the  surveyor  who  uses  this  instrument.  The  limits  of  our 
space  will  allow  only  a  brief  reference  to  its  principal  features. 
Fig.  15  exhibits  the  instrument  by  itself;  Fig.  16,  p.  31,  is  a 
graphical  illustration  of  the  solar  apparatus  as  an  attachment  to 
the  transit,  the  circles  shown  being  intended  to  represent  those 
supposed  to  be  drawn  upon  the  concave  surface  of  the  heavens. 
The  form  of  the  solar  compass  shown  in  Fig.  15  has  the 
arrangement  of  its  sockets  and  plates  similar  to  that  of  the 
transit,  the  standards  similar  to  those  of  the  compass,  the  solar 
apparatus  being  placed  on  the  upper  vernier  plate  and  taking 
the  place  of  the  needle,  for  which  it  operates  as  a  substitute 
in  the  field. 

The  solar  compass  consists  mainly  of  three  arcs  of  circles,  a 
the  latitude  arc,  by  which  is  set  off  the  latitude  of  the  place,  b 
the  declination  arc,  by  which  is  set  off  the  declination  of  the 
sun,  and  c  the  hour  arc,  by  which  is  set  off  the  hour  of  the  day. 
The  arm  h  is  movable  about  a  point  at  the  extremity  of  the 
piece  containing  the  declination  arc,  there  being  at  each  end  a 
solar  lens  having  its  focus  on  a  silvered  plate  on  the  other  end. 
The  arc  of  the  declination  limb  turns  on  an  axis,  and  one  or 
the  other  solar  lens  is  used,  according  as  the  sun  is  north  or 
south  of  the  equator.  Fig.  15  shows  the  position  of  the  decli- 
nation arc  when  the  sun  is  south;  Fig.  16,  when  it  is  north. 
The  needle  box  is  moved  about  its  centre  by  a  slow-motion 
screw.  It  contains  a  magnetic  needle,  and  is  furnished  with  a 
graduated  arc  about  36°  in  extent. 


FIELD   INSTRUMENTS 


29 


FIG.  15.    BUBT'S  SOLAR  COMPASS 


FIELD   INSTRUMENTS 


31 


FIG.  16.    TRANSIT  WITH  SOLAR  ATTACHMENT 

The  circles  shown  in  the  cut  are  intended  to  represent  in  miniature  circles  supposed 
to  be  drawn  upon  the  concave  surface  of  the  heavens. 


32  SURVEYING 

The  solar  compass  may  be  used  for  most  of  the  purposes  of  a 
compass  or  transit.  Its  most  important  use,  however,  is  to  run 
north  and  south  lines,  especially  in  laying  out  the  public  lands. 
It  may  be  used  also  in  determining  the  latitude  of  a  place. 

To  establish  a  True  Meridian.  Set  off  on  the  latitude  arc 
the  latitude  of  the  place.  Set  off  on  the  declination  arc  the 
declination  of  the  sun,  corrected  for  refraction.  Set  the 
instrument  over  the  station ;  level,  and  turn  the  sights  in  a 
north  and  south  direction  by  the  needle.  The  surveyor  then 
turns  the  solar  lens  to  the  sun,  and  with  one  hand  on  the 
instrument  and  the  other  on  the  revolving  arm,  moves  both 
from  side  to  side  until  the  sun's  image  is  made  to  appear  on 
the  silvered  plate,  precisely  between  the  equatorial  lines.  The 
line  of  sights  then  indicates  the  true  meridian. 

The  bearing  of  any  line  from  the  meridian  may  be  read  by 
the  verniers  of  the  horizontal  limb.  When  a  due  east  and 
west  line  is  to  be  run,  these  verniers  are  set  at  90°,  and  the 
sun's  image  is  kept  between  the  lines  as  before. 

Other  Methods.  By  North  Star  at  Culmination.  The  North 
Star,  or  Polaris,  at  present  revolves  about  the  north  pole  of 
the  heavens  at  the  distance  of  about  1^° ;  hence,  it  is  on  the 
meridian  twice  in  23  h.  56  m.  4  s.  (a  sidereal  day),  once  above 
the  pole,  called  the  upper  culmination,  and  11  h.  58  m.  2  s. 
later  below  the  pole,  called  the  lower  culmination. 

The  time  of  the  upper  culmination  of  Polaris  may  be  found 
by  means  of  the  star  Mizar,  the  middle  one  of  the  three  stars 
in  the  handle  of  the  Dipper  (in  the  constellation  of  the  Great 
Bear).  It  crosses  the  meridian  at  nearly  the  same  time  as 
Polaris.  Suspend  a  plumb  line,  placing  the  bob  in  a  pail  of 
water  to  lessen  its  vibrations.  South  of  the  plumb  line,  upon 
a  horizontal  board  firmly  supported,  place  a  compass  sight,  or 
any  upright  with  a  small  opening  or  slit,  fastened  to  a  board 
a  few  inches  square.  At  night,  when  Mizar  by  estimation 
approaches  the  meridian,  place  the  compass  sight  in  line  with 


FIELD   INSTRUMENTS 


33 


Cassiopeiae 


Polaris 


Polaris  and  the  plumb  line,  and  move  it  so  as  to  keep  it  in 
this  line  until  the  plumb  line  falls  also  on  Mizar  (Fig.  17). 
Note  the  time ;  then  (1903)  3  in.  39  s.  later  Polaris  will  be 
on  the  meridian.  If  then  Polaris,  the  plumb  line  and  the  com- 
pass sight  are  brought  into  line,  the  plumb  line  and  compass 
sight  will  give  two  points  in  the  meridian  ;  or  if  the  telescope 
of  the  transit  is  brought  to  bear  on  Polaris, 
and  a  light  is  held  near  to  make  the  wires 
visible  if  necessary,  the  telescope  will 
then  lie  in  the  plane  of  the  meridian, 
which  may  be  marked  by  bringing  the 
telescope  to  a  horizontal  position. 

For  each  year  subsequent  to  1903  add 
21  s.  to  3  m.  39  s.  If  the  lower  culmina- 
tion takes  place  at  night,  the  time  may  be 
found  in  a  similar  manner.  When  Mizar 
cannot  conveniently  be  used,  8  Cassiopeiae 
(Fig.  17)  may  be  employed,  the  method 
being  the  same  as  in  the  case  of  Mizar. 
The  interval,  however  (1903),  is  4  m.  24  s. 
and  the  annual  increase  of  the  interval  Mizar  ±  * 
about  20  s. 

By  North  Star  at  Greatest  Elongation. 
When  Polaris  is  at  its  greatest  apparent 
angular  distance  east  or  west  of  the  pole, 
it  is  said  to  be  at  greatest  elongation.  It 
attains  its  greatest  eastern  elongation  and 
western  elongation,  respectively,  5  h.  59  m.  1  s.  after  lower  and 
upper  culmination.  The  azimuth  of  a  star  is  the  angle  which 
the  meridian  plane  makes  with  the  vertical  circle  passing 
through  the  star  and  the  zenith  of  the  observer. 

If  now  we  know  the  time  of  either  extreme  elongation  and 
also  the  azimuth  of  Polaris  at  an  extreme  elongation,  we  can 
from  these  data  establish  a  true  meridian.  The  latter  of  these 


Pole 


Great 


Bear 


FIG.  17 


34  SURVEYING 

data  is  given  for  various  latitudes  and  for  years  to  come  in 
tables,  to  which  the  surveyor  is  supposed  to  have  access.  To 
obtain  a  line  in  the  direction  of  Polaris  at  greatest  elongation, 
we  may  proceed  as  follows :  A  few  minutes  before  the  time 
of  greatest  elongation,  place  the  compass  sight  in  line  with 
the  plumb  line  and  Polaris,  keeping  it  in  line  with  these  until 
the  star  begins  to  'recede.  At  this  moment  the  sight  and 
plumb  line  are  in  the  required  line.  Or  bring  the  telescope 
of  the  transit  to  bear  on  the  star,  and  follow  it  keeping  the 
vertical  wire  over  the  star  until  it  begins  to  recede.  The 
telescope  will  then  be  in  the  required  line.  In  either  case, 
after  having  the  transit  sighted  in  the  direction  of  the  line 
just  found,  turn  it  in  the  proper  direction  through  an  angle 
equal  to  the  azimuth  as  found  from  the  tables. 

The  accompanying  table  *  gives  the  Washington  mean  time 
of  each  tenth  transit  of  Polaris  (upper  culmination)  at  the 
meridian  of  Washington,  D.C.  The  last  column  contains  the 
variation  per  day,  to  facilitate  the  interpolation  of  the  time 
for  any  intermediate  transit. 

The  transit  which  occurs  October  17  is  the  tenth  transit 
following  that  which  occurs  on  October  8.  This  is  because 
two  transits  occur  on  October  13  ;  the  interval  separating  them 
being  23  h.  56  m.  4  s.  of  mean  time.  These  two  transits  are 
introduced  in  the  table  for  greater  convenience,  and  as  a  safe- 
guard against  error  respecting  the  particular  day  of  transits  in 
that  vicinity.  The  double  lines  merely  call  attention  to  the 
break  thus  caused  in  the  series. 

By  interpolation  we  may,  by  taking  account  of  the  longitude 
of  any  given  station,  find  the  local  mean  time  of  transit  of 
Polaris  at  that  station  for  any  particular  day.  Thus,  to  find 
the  Cincinnati  mean  time  of  the  upper  culmination  of  Polaris 
at  Cincinnati,  on  May  15,  1902,  we  have  (p.  36)  : 

*  Furnished  by  the  Director  of  the  Nautical  Almanac  Office,  Wash- 
ington, D.C. 


FIELD   INSTRUMENTS 


35 


DAY  OF 

LOCAL  MEAN  TIME 

CHANGE  IN 

THE  YEAR 

OF  EVERY  lOrn  TRANSIT 

1  DAY 

1902 

(Civil  Time.) 

Jan.   1 

6h  41m 

19s  P.M. 

-  3m  56s.  8 

11 

6   1 

51  " 

56.9 

21 

5  22 

22   '  ' 

56.9 

31 

4  42 

53   ' 

56.9 

Feb.  10 

4   3 

24   ' 

56.8 

20 

3  23 

56   ' 

56.7 

Mar.  2 

2  44 

29   ' 

56.6 

12 

2   5 

4  " 

56.4 

22 

1  25 

41  " 

56.2 

Apr.   1 

12  46 

20  " 

56.0 

11 

12   7 

1  " 

55.8 

21 

11  27 

44  A.M. 

55.7 

May  1 

10  48 

28  " 

55.7 

11 

10.  9 

14  " 

55.3 

21 

9  30 

2  " 

55.1 

31 

8  50 

51   ' 

55.0 

June  10 

8  11 

41   ' 

55.0 

20 

7  32 

31   ' 

54.9 

30 

6  53 

22   ' 

54.9 

July  10 

6  14 

13   ' 

54.8 

20 

5  35 

5   ' 

54.9 

30 

4  55 

56   ' 

54.9 

Aug.  9 

4  16 

47   ' 

55.0 

10 

3  37 

37   ' 

55.1 

29 

2  58 

26   ' 

55.2 

Sept.  8 

2  19 

14   ' 

55.3 

18 

1  40 

1   ' 

55.4 

28 

1   0 

46   ' 

55.5 

Oct.   8 

12  21 

30   ' 

55.7 

Oct.  13 

12   1 

51  A.M. 

Oct.  13 

11  57 

55  P.M. 

Oct.  17 

11  42 

12  P.M. 

55.8 

27 

11   2 

53  " 

56.0 

Nov.  6 

10  23 

32  " 

56.1 

16 

9  44 

10  " 

56.3 

26 

9  4 

46  " 

56.4 

Dec.  6 

8  25 

21  " 

56.5 

16 

7  45 

55  " 

56.7 

26 

7  6 

27  " 

56.8 

36 

6  26 

58  " 

-3  56.9 

36  SURVEYING 

Local  mean  time  of  transit  at  Washington,  May  11,  1902 

=  10h  9m  14s  A.M. 
Longitude  of  Cincinnati  west  of  Washington 

=  +  Oh  29m  40s  =  +  Od.021. 
May  15d  +  Od.021  =  May  15d.021. 
Preceding  tabular  date      =  May  11. 
Therefore,  interval  =  4d.021. 

Daily  variation  =  -  3m  558.3  =  -  2358.3. 

Total  change  =  4.021  x  (-  2358.3)  =  -  15m  46s. 

10h        9m  14s  A.M. 

-15     46 
9h      53m  28s  A.M. 

Therefore,  the  required  time  is  9h.53m  28s  A.M.,  May  15, 1902. 


SECTION  YII 
THE   Y  LEVEL 

Description.  The  essential  parts  of  the  Y  level  (Fig.  18) 
are,  the  telescope,  which  is  of  various  lengths,  usually  about 
20  inches,  and  rests  on  supports  called  Y's,  from  their  shape ; 
the  spirit  level,  which  is  under  the  telescope  and  attached  to 
it ;  and  the  leveling  head  and  tripod,  which  are  similar  to  the 
same  parts  of  the  transit. 

Leveling  Rod.  There  are  several  kinds  of  leveling  rods, 
each  possessing  some  merit  peculiar  to  its  purpose.  The  one 
shown  in  Fig.  19  is  known  as  the  Philadelphia  leveling  rod, 
and  is  the  one  in  most  common  use.  It  is  made  of  two  pieces 
of  wood,  sliding  upon  each  other,  and  held  in  position  by  a 
clamp.  The  front  surface  of  each  piece  is  graduated  to  hun- 
dredths  of  a  foot  up  to  7  feet ;  the  back  surface  of  the  real- 
piece  is  figured  from  7  to  13  feet,  reading  from  the  top  down, 


FIELD   INSTRUMENTS 


37 


FIG.  18.    THE  Y  LEVEL 


38  SURVEYING 

and  it  also  has  a  scale  by  which  the  rod  is  read  to  half  hun- 
dredths  of  a  foot  as  it  is  extended.  The  target 
slides  along  the  front  of  the  rod  and  is  held  in 
place  by  two  springs  which  press  upon  the  sides 
of  the  rod.  It  has  a  square  opening  at  the  centre, 
through  which  the  division  line  of  the  rod  oppo- 
site to  the  horizontal  line  of  the  target  may  be 
seen.  It  also  carries  a  scale  by  which  heights 
may  be  read  to  half  hundredths  of  a  foot.  For 
heights  not  greater  than  7  feet,  the  target  is 
moved  up  or  down  the  front  surface,  the  rod 
being  closed  and  clamped;  but  when  a  greater 
height  i-s  required  the  target  is  fixed  at  7  feet 
and  the  rear  half  of  the  rod  extended  to  the 
required  height.  The  rod  thus  becomes  a  self- 
reading  rod  13  feet  long. 

How  to  use  Level  and  Rod.  When  the  leveling 
instrument  is  used,  the  tripod  should  be  set  firm ; 
the  spirit  level  should  then  be  brought  succes- 
sively over  each  opposite  pair  of  leveling  screws 
and  leveled  in  each  position,  the  operation  being 
repeated  until  the  bubble  remains  in  the  middle 
of  the  tube  through  an  entire  rotation  of  the  tele- 
scope. Each  time  before  taking  an  observation 
the  instrument  should  be  examined  to  see  if  it 
is  still  level.  Care  should  be  taken  to  bring  the 
cross  wires  of  the  telescope  precisely  in  focus  and 
the  object  into  such  perfect  view  that  the  wires 
will  appear  to  be  fastened  to  the  surface,  how- 
ever the  eye  is  moved.  For  very  accurate  work 
the  instrument  should  be  shielded  from  the  direct 
rays  of  the  sun. 
FIG.  19  The  leveling  rod  should  be  held  in  a  truly 

vertical  position,  the  rodman    standing   squarely  behind  it. 


FIELD   INSTRUMENTS 


39 


The  target  is  then  raised  or  lowered  at  the  signal  of  the 
leveler  until  its  horizontal  line  is  cut  by  the  intersection  of 
the  cross  wires  of  the  telescope.  The  reading  is  done  by  the 
leveler  or  the  rodinan  according  to  the  kind  of  rod  used. 

Substitutes  for  the  Y  Level.  For  ordinary  work,  the  Sur- 
veyor's or  Engineer's  Transit  is  often  used. 

The  plumb  level  (Fig.  20)  consists  of  two  pieces  of  wood 
joined  at  right  angles.  A  straight  line  is  drawn  on  the  upright 
perpendicular  to  the  upper  edge  of  the  crosshead.  The  instru- 
ment is  fastened  to  a  support  by  a  screw  through  the  centre  of 
the  crosshead.  The  upper  edge  of  the  crosshead  is  brought 
to  $>  level  by  making  the  line  on  the  upright  coincide  with  a 
plumb  line. 


n 


9 

FIG.  20 


FIG.  21 


FIG. 


A  carpenter's  square  can  be  made  into  a  level  by  being 
supported  by  a  post  (Fig.  21),  the  top  of  which  is  split  or  sawed 
so  as  to  receive  the  longer  arm.  The  shorter  arm  is  made 
vertical  by  a  plumb  line,  which  brings  the  longer  arm  to  a 
level. 

Tlie  water  level,  as  shown  in  Fig.  22,  consists  of  two 
upright  glass  tubes  cemented  into  a  connecting  tube  of  any 
material.  The  whole  is  nearly  filled  with  water  and  sup- 
ported at  a  convenient  height.  The  surface  of  the  water 
in  the  uprights  determines  the  level.  The  water  should  be 
colored. 


40  SURVEYING 

A  level  line  may  be  obtained  by  sighting  along  the  upper 
surface  of  the  block  in  which  an  ordinary  spirit  level  is 
mounted. 

For  many  purposes  not  requiring  great  accuracy,  any  of  the 
foregoing  simple  instruments  in  connection  with  any  graduated 
rod  will  be  sufficient. 


EXERCISE   IV 

1.  Set  up  the  level  and  take  the  readings  on  the  leveling 
rod  at  two  stations  equally  distant  from  the  instrument.    What 
does  the  difference  of  these  readings  indicate  ? 

2.  Set   up   the    level    successively  at  the  two  stations  in 
Example  1,  taking  the  readings  on  the  leveling  rod  placed 
where  the  instrument  was  first.     What  does  the  difference  of 
these   readings   indicate?     Ought  this  difference  to  be  the 
same  as  that  in  Example  1  ?     Explain. 

3.  In  the  field  of  Example  1,  Exercise  II,  p.  22,  set  up  the 
level  successively  at  the  middle  of  each  of  the  five  sides,  taking 
the  readings  on  the  rod  each  time  at  both  adjacent  stations  of 
the  field.     Find  the  difference  between  the  sum  of  the  hind- 
sights and  the  sum   of   the   foresights.    What   should   this 
difference  equal? 


SECTION  VIII 
THE   PLANE   TABLE 

Description  and  Uses.  The  plane  table,  an  approved  form  of 
which  is  shown  in  Fig.  23,  consists  mainly  of  a  drawing 
board  made  of  well-seasoned  wood,  arranged  in  sections  to  pre- 
vent warping,  and  supported  at  a  convenient  height  by  a  tripod 
and  leveling  head,  with  attachments  for  horizontal  movement. 


FIELD   INSTRUMENTS 


41 


FIG.  23.    THE  PLANE  TABLE 


42  SURVEYING 

The  board  is  provided  with  rollers  or  clamps  or  both,  for 
keeping  the  paper  secure  and  even.  The  plumbing  arm  has 
its  end  brought  to  a  point  which,  however  placed  on  the 
paper,  is  directly  above  the  corresponding  point  on  the  ground 
determined  by  the  plummet.  The  alidade  is  a  ruler  of  brass 
or  steel  supporting  a  telescope  with  stadia  or  sight  standards, 
whose  line  of  sight  is  in  or  parallel  to  the  same  vertical  plane 
with  the  beveled  edge  of  the  ruler.  A  compass  with  two 
spirit  levels  serves  both  to  level  the  table  and,  when  applied 
by  the  edges  parallel  to  the  zero,  line  of  the  compass  circle, 
to  determine  the  magnetic  bearing  of  the  lines  drawn  on  the 
paper,  or  the  direction  of  the  table  itself. 

After  the  principal  lines  of  a  survey  have  been  determined 
and  plotted,  the  details  of  the  plot  may  be  filled  in  by  means 
of  the  plane  table ;  or,  when  a  plot  only  of  a  tract  of  land  is 
desired  and  extreme  accuracy  is  not  required,  this  instrument 
affords  the  most  expeditious  means  of  obtaining  it.  There  is 
little  use  for  it  outside  of  the  United  States  Coast  and  Geodetic 
Survey  and  the  United  States  Geological  Survey. 

To  orient  the  Table.  This  operation  consists  in  placing  the 
table  so  that  the  lines  of  the  plot  shall  be  parallel  to  the 
corresponding  lines  on  the  ground. 

This  may  be  accomplished  approximately  by  turning  the 
table  until  the  needle  of  the  compass  indicates  the  same  bear- 
ing as  at  a  previous  station,  the  edge  of  the  compass  coinciding 
with  the  same  line  on  the  paper  at  both  stations. 

If,  however,  the  line  connecting  the  station  at  which  the 
instrument  is  placed  with  another  station  is  already  plotted, 
the  table  may  be  placed  in  position  accurately  by  placing  it 
over  the  station  so  that  the  plotted  line  is  by  estimation  over 
and  in  the  direction  of  the  line  on  the  ground ;  then  making 
the  edge  of  the  ruler  coincide  with  the  plotted  line,  and  turn- 
ing the  board  until  the  line  of  sight  bisects  the  signal  at  the 
other  end  of  the  line  on  the  ground. 


FIELD   INSTRUMENTS 


43 


To  plot  any  Point.  Let  ab  on  the  paper  represent  the  line 
AB  on  the  ground;  it  is  required  to  plot  c,  representing  C  on 
the  ground. 

1.  By  intersection. 

Place  the  table  in  position  at  A  (Fig.  24),  plumbing  a  over  A,  and 
making  the  fiducial  edge  of  the 
ruler  pass  through  a;   turn  the  •]$ 

alidade  about  a  until  the  line  of  /      VN 

sight  bisects  the  signal  at  C,  and  j  X 

draw  a  line  along  the  fiducial  edge 
of  the  ruler.  Place  the  table  in 
position  at  5,  plumbing  b  over  B, 
and  repeat  the  operation  just 
described.  Then  c  is  the  intersec- 
tion of  the  two  lines  thus  drawn. 

2.  By  resection. 

Place  the  table  in  position  at  A  (Fig.  25),  and  draw  a  line  in  the  direc- 
tion of  (7,  as  in  the  former  case  ;  then  remove  the  instrument  to  C,  place 

it  in  position  by  the  line  drawn 
from  a,  make  the  edge  of  the 
ruler  pass  through  &,  and  turn 
the  alidade  about  b  until  B  is  in 
the  line  of  sight.  A  line  drawn 
along  the  edge  of  the  ruler  will 
intersect  the  line  from  a  in  c. 

3.  By  radiation. 

Place  the  table  in  position  at  A 
(Fig.  26),  and  draw  a  line  from  a 
toward  C,  as  in  the  former  cases. 
Measure  AC,  and  lay  off  ac  to 
the  same  scale  as  ab. 

To  plot  a  Field  AB  CD  ••• 

By  radiation. 

Set  up  the  table  at  any  point 
P,  and  mark  p  on  the  paper  over 
FIG.  26  P.     Draw  indefinite  lines  from  p 


44 


SURVEYING 


toward  J.,  B,  C,  •  •  •     Measure  PA,  PJB,  •  •  • ,  and  lay  off  pa, 
suitable  scale,  and  join  a  and  6,  6  and  c,  c  and  d,  •  •  • 


to  a 


By  progression. 

Set  up  the  table  at  A,  and  draw 
a  line  from  a  toward  B.  Measure 
AB,  and  plot  ab  to  a  suitable 
scale.  Set  up  the  table  in  position 
at  B,  and  in  like  manner  deter- 
mine and  plot  be  ;  and  so  on. 


FIG.  27 


By  intersection. 

Plot  one  side  as  a  base  line. 

Plot  the  other  corners  by  the  method  of  intersection,  and  join  these  points 
in  proper  order  by  straight  lines. 

By  resection. 

Plot  one  side  as  a  base  line.     Plot  the  other  corners  by  the  method  of 
resection,  and  join  these  points  in  proper  order  by  straight  lines. 

The  Three-Point  Problem.     Let  A,  B,  C  represent  three  field 
stations  plotted  as  a,  5,  c,  respectively  (Fig.  28) ;  it  is  required 


FIG.  28 


to  plot  d  representing  a  fourth  field  station  D,  from  which  A, 
B,  and  C  are  visible. 

Place  the  table  over  D,  level  and  orient  approximately  by 
the  compass.     Determine  d  by  resection  as  follows :  Make  the 


FIELD   INSTRUMENTS  45 

edge  of  the  ruler  pass  through  a  and  lie  in  the  direction  a  A, 
and  draw  a  line  along  the  edge  of  the  ruler.  In  like  manner, 
draw  lines  through  b  toward  B  and  through  c  toward  C.  If 
the  table  is  oriented  perfectly,  these  lines  meet  at  the  required 
point  d,  but  ordinarily  they  will  form  the  triangle  of  error,  ab, 
ac,  be.  In  this  case,  through  a,  b,  and  ab ;  a,  c,  and  ac ;  and 
b,  c,  and  be,  respectively,  draw  circles ;  these  circles  will  inter- 
sect in  the  required  point  d.  For  at  the  required  point  the 
sides  ab,  ac,  be  must  subtend  the  same  angle  as  at  the  points 
ab,  ac,  be,  respectively.  Hence,  the  required  point  d  lies  at 
the  intersection  of  the  three  circles  mentioned.  The  plane 
table  may  now  be  oriented  accurately. 

The  three-point  problem  may  also  be  solved  by  fastening  on  the  board 
a  piece  of  tracing  paper  and  marking  the  point  d  representing  D,  after 
which  lines  are  drawn  from  d  toward  A,  B,  and  C.  The  tracing  paper 
is  then  moved  until  the  lines  thus  drawn  pass  through  a,  6,  c.  respectively, 
when  by  pricking  through  d  the  point  is  determined  on  the  plot  below. 
This  method,  however,  is  impracticable  in  case  the  wind  blows. 


CHAPTER   II 
OFFICE   INSTRUMENTS 

SECTION   IX 
PLOTTING   INSTRUMENTS 

Definitions.  A  map  is  a  representation  by  means  of  points, 
lines,  and  conventional  signs  on  a  plane  surface,  as  on  paper 
of  a  surveyed  portion  of  the  earth's  surface,  including  objects 
upon  it.  If  only  the  boundary  lines  are  drawn,  the  repre- 
sentation is  called  an  outline  map,  or  plot.  The  plot  is  a 
figure  similar  to  the  original,  and  the  ratio  of  a  line  of  the 
field  to  the  corresponding  line  of  the  plot  is  called  the  scale. 
In  surveying  it  is  customary  to  designate  the  scale  as  so  many 
chains  to  the  inch. 

Principal  Minor  Instruments.  The  principal  minor  instru- 
ments used  in  plotting  are  a  ruler,  pencil,  straight-line  pen, 
hair-spring  dividers,  compasses,  a  right  triangle  of  wood  or 
hard  rubber,  a  T-square,  and  a  parallel  ruler. 

The  Diagonal  Scale.  A  portion  of  this  scale  is  shown  in 
Fig.  29.  AB  is  the  unit.  AB  and  A'B'  are  divided  into  ten 
equal  parts,  and  B  is  joined  with  h,  the  first  division  point  to 
the  left  of  B';  the  first  division  point  to  the  left  of  B  is  joined 
with  the  second  to  the  left  of  B',  and  so  on.  The  part  of  the 
horizontal  line  numbered  1  intercepted  between  BB'  and  Bh  is 
evidently  y1^  of  y1^  =  T^  of  the  unit ;  the  part  of  the  hori- 
zontal line  numbered  2  intercepted  between  BB'  and  Bh  is 

of  the  unit,  and  so  on. 

46 


OFFICE   INSTRUMENTS 


47 


The  method  of  using  this  scale  is  as  follows : 
Let  it  be  required  to  lay  off  the  distance  1.43. 


A' 


10  | 


h   B' 


C' 


10    9 

A 


o 
D 

FIG.  29 


C 


Place  one  foot  of  the  dividers  at  the  intersection  of  the  horizontal  line 
numbered  3  and  the  diagonal  numbered  4,  and  place  the  other  foot  at 
the  intersection  of  the  vertical  line  numbered  1  (CC")  and  the  horizontal 
line  numbered  3  ;  the  distance  between  the  feet  of  the  dividers  will  be 
the  distance  required.  For,  measuring  along  the  horizontal  line  num- 
bered 3,  from  CC"  to  BB'  is  1 ;  from  BB'  to  Bh  is  0.03  ;  and  from  Bh  ta 
the  diagonal  numbered  4  is  0.4  ;  and  1  +  0.03  +  0.4  =  1.43. 

The  Circular  Protractor.  This  instrument  (Fig.  30)  usually 
consists  of  a  semicircular  piece  of  brass  or  german  silver,  with 
its  arc  divided  into  degrees  and  its  centre  marked. 

Some  protractors  have  an  arm  which  carries  a  vernier,  loy 
which  angles  may  be  constructed  to  single  minutes.  Still 
others  embrace  an  entire  circle  and  have  several  arms  with 
verniers. 

A  rectangular  protractor,  having  the  degrees  marked  off  on 
three  sides  of  a  plane  scale,  is  sometimes  used.  Often  this 
form  of  the  protractor  is  found  on  the  reverse  side  of  the 
diagonal  scale. 


48  SURVEYING 

Constructions.  1.  To  lay  off  an  angle  with  the  circular 
protractor.  Place  the  centre  over  the  vertex  of  the  angle, 
and  make  the  diameter  coincide  with  the  given  side  of  the 
angle.  Mark  off  the  number  of  degrees  in  the  given  angle, 
and  draw  a  line  through  this  point  and  the  vertex. 


FIG.  so 

2.  To  draw  through  a  given  point  a  line  parallel  to  a  given 
line  with  a  right  triangle  and  ruler. 

Make  one  of  the  sides  of  the  triangle  coincide  with  the 
given  line,  and,  placing  the  ruler  against  one  of  the  other  sides, 
move  the  triangle  along  the  ruler  until  the  first  side  passes 
through  the  given  point ;  then  draw  a  line  along  this  side. 

3.  To  draw  through  a  given  point  a  line  perpendicular  to 
a  given  line  with  a  right  triangle  and  ruler. 

Make  the  hypotenuse  of  the  right  triangle  coincide  with 
the  given  line,  and,  placing  a  ruler  against  one  of  the  other 
sides  of  the  triangle,  revolve  the  triangle  about  the  vertex  of 
the  right  angle  as  a  centre  until  its  other  perpendicular  side 
is  against  the  ruler ;  then  move  the  triangle  along  the  ruler 
until  the  hypotenuse  passes  through  the  given  point,  and 
draw  a  line  along  the  hypotenuse. 


OFFICE   INSTRUMENTS  49 

SECTION   X 
COMPUTING  INSTRUMENTS 

The  Planimeter.  This  is  an  instrument  for  measuring  the 
area  of  any  irregular  field,  by  applying  it  to  a  plot  of  the 
field  drawn  accurately  to  scale.  The  form  in  most  common 
use  is  that  known  as  the  polar  planimeter.  The  essential 
parts  are  two  arms,  one  fixed  in  length,  the  other  adjustable, 
and  a  rolling  wheel  mounted  on  an  axis  parallel  to  the  adjust- 
able arm.  The  outer  end  of  the  arm  of  fixed  length  is  made 
fast  to  the  plot  by  means  of  a  needle  point,  and  the  free  end 
of  the  other  arm  is  made  to  trace  the  perimeter  of  the  figure 
to  be  measured.  A  disk  records  the  area  in  the  unit  for 
which  the  instrument  is  set. 

The  Slide  Rule.  This  is  an  instrument  for  effecting  the 
processes  of  multiplication,  division,  involution,  and  evolution 
by  means  of  logarithms.  It  consists  of  a  series  of  scales  so 
arranged  that  by  sliding  one  upon  the  other  the  addition  or 
subtraction  of  logarithms  is  mechanically  performed.  For  a 
full  description  of  this  labor-saving  device  in  its  various  forms, 
the  student  is  referred  to  some  treatise  on  the  subject. 


CHAPTER   III 
LAND   SURVEYING 

SECTION   XI 
DEFINITIONS 

Land  Surveying  is  the  art  of  measuring,  laying  out,  and 
dividing  land,  computing  parts  and  areas  from  measured  parts, 
and  preparing  a  plot.  An  original  survey  includes  laying  out 
the  boundary  lines  and  establishing  the  corners.  A  resurvey 
is  the  retracing  of  old  boundary  lines  and  the  finding  of  corner 
monuments,  or  the  relocating  of  them  when  lost. 
Rules  for  Areas.  The  unit  of  land  measure  is  the 

acre  =  10  square  chains  =  4  roods 

=  160  square  rods,  perches,  or  poles. 

Areas  are  referred  to  the  horizontal  plane,  no  allowance 
being  made  for  inequalities  of  surface. 

Let  .4,  B,  and  C  be  the  angles  of  a  triangle,  and  a,  £,  and 
c  the  opposite  bides,  respectively,  and  let  s  =  %(a  +  b  -j-  c). 
Area  of  triangle  ABC  —  £  base  x  altitude 
=  |-  be  sin  A 

a2  sin  B  sin  C 
~  2  sin  (B  +  C) 
=  Vs  (s  —  a)  (s  —  b)  (s  —  c). 
Area  of  rectangle  =  base  x  altitude. 
Area  of  trapezoid  =  £  sum  of  parallel  sides  x  altitude. 

NOTE.     Spanish  American  units  are  in  use  in  Texas,  California,  and 
Mexico.     In  this  system  the  vara  is  the  unit  of  length,  which  in  Texas  is 

50 


LAND    SURVEYING  51 

reckoned  33£  inches,  in  California  33  inches,  in  Mexico  32.9927  inches. 
The  area  of  a  square  1000  varas  on  a  side  is  called  a  labor,  and  of  a 
square  5000  varas  on  a  side  is  called  a  league. 


SECTION   XII 
SPECIAL    METHODS    OF    SURVEYING,    AND    COMPUTING    AREAS 

Triangular  Fields.  Measure,  as  may  be  most  convenient, 
the  three  sides,  two  sides  and  the  included  angle,  two  angles 
and  the  included  side,  or  a  side  and  the  altitude  upon  that 
side,  and  compute  the  area  by  the  appropriate  formula. 

Fields  having  More  than  Three  Straight  Sides.  Divide  the 
field  into  triangles  and  take  the  sum  of  the  areas  of  the 
triangles.  Or,  run  a  diagonal  and  perpendiculars  to  it  from 
the  opposite  vertices ;  take  the  sum  of  the  areas  of  the  right 
triangles,  rectangles,  and  trapezoids  thus  formed. 

A  third  method  is  as  follows:  Let  ABCD  (Fig.  31)  repre- 
sent a  field,  and  P  and  P'  two  stations  within  it.  (They  may 
be  without  the  field.)  Measure  D 
PP'  with  great  exactness.  Meas- 
ure the  angles  between  PP'  and 
the  lines  from  P  and  P'  to  the 
corners  of  the  field. 

In  the  triangle  P'PD,  PP'  and 
the  angles  PP'D  and  P'PD  are 
known ;  hence,  PD  may  be  found. 
In  like  manner,  PC  may  be  found. 
Then,  in  the  triangle  PDC,  PD, 
PC,  and  the  angle  DPC  are  known ;  hence,  the  area  of  PDC 
may  be  computed.  In  like  manner,  the  areas  of  all  the  trian- 
gles about  P  or  P'  may  be  determined. 

Area  ABCD  =  PAD  +  PDC  +  PCB  +  PEA', 
also,        area   ABCD  =  P'AD  +  P'DC  +  P'CB  +  P'BA. 


52 


SURVEYING 


Fields  having  Irregular  Boundary  Lines.  Let  A  GBCD  (Fig.  32) 
represent  a  field  having  a  stream  AEFGHKB  as  a  boundary 
line.  Run  the  line  AB.  From  E,  F,  G,  H,  K,  prominent 
points  on  the  bank  of  the  stream,  let  fall  perpendiculars  EE', 
FF',  GG'9  etc.,  upon  AB.  Regarding  AE,  EF,  etc.,  as  straight 


FIG.  32 


FIG.  33 


lines,  the  portion  of  the  field  cut  off  by  AB  is  divided  into 
right  triangles,  rectangles,  and  trapezoids,  the  necessary  ele- 
ments of  which  can  be  measured  and  the  areas  computed. 
The  sum  of  these  areas  added  to  the  area  of  A  BCD  gives 
the  area  required.  If  the  offsets  are  at  regular  intervals,  then 
the  area  of  the  part  cut  off  by  AB  may  be  found  by  adding  the 
offsets  and  multiplying  by  the  common  distance  between  them. 

When  the  irregular  boundary  line  crosses  the  straight  line 
that  joins  its  extremities,  as  in  Fig.  33,  the  areas  of  £EFH  and 
HGB  may  be  found  separately,  as  in  the  preceding  case.  Then, 
the  area  required  =  ABCD  +  HGB  —  AEFH. 

Rectangular  System  of  Co-ordinates.  Let  XX'  and  YY'  (Fig. 
34)  be  two  fixed  perpendicular  lines  intersecting  at  the  point  0. 
Let  the  four  parts  into  which  these  lines  divide  the  plane  be 
called  Quadrants,  as  in  Trigonometry,  and  be  distinguished  by 
naming  them,  respectively,  first,  second,  third,  'and  fourth 
quadrants. 

Suppose  the  position  of  a  point  is  described  by  saying  that 
its  distance  from  YY',  expressed  in  terms  of  some  chosen  unit 


LAND   SURVEYING  53 

of  length,  is  3,  and  its  distance  from  XX'  is  4.  Then  there 
is  in  each  quadrant  one  point  and  only  one  which  will 
satisfy  these  conditions.  The  position  of  the  point  in  each 
quadrant  may  be  found  by  drawing  parallels  to  YY'  at  the 
distance  3  from  YY',  and  parallels  -^D 
to  XX'  at  the  distance  4  from  XX' ;  E  \p 
then  the  intersections  Plt  P2,  P3, 
and  P4  satisfy  the  given  condi- 
tions. X- rr 


O 

In   order   to   determine   which  ] 


IV 


-H 


one  of  the  four  points,  P1?  P2,  P3, 

P4,  is  meant,  we  adopt   the  rule       O- 

that  distances  measured  from  YY1  &          Y/ 

to  the  right  are  positive;    to  the 

left,   negative.      Distances   measured  from  XX'  upward   are 

positive  ;  downward,  negative.     Then,  the  position  of  Px  will 

be  denoted  by  +  3,  +  4 ;  of  P2,  by  —  3,  +  4  ;  of  P8,  by  —  3,  —  4; 

of  P4,  by  -f  3,  -  4. 

The  fixed  lines  XX'  and  YY'  are  called  the  Axes  of  Co-ordi- 
nates ;  XX'  is  called  the  Axis  of  Abscissas,  or  Axis  of  x ;  YY', 
the  Axis  of  Ordinates,  or  Axis  of  y.  The  intersection  0  is 
called  the  Origin. 

The  two  distances  (with  signs  prefixed)  which  determine 
the  position  of  a  point  are  called  the  Co-ordinates  of  the 
point ;  the  distance  of  the  point  from  YY1  is  called  its 
Abscissa  ;  and  the  distance  from  XX',  its  Ordinate. 

Abscissas  are  usually  denoted  by  x,  and.  ordinates  by  y, 
and  a  point  is  represented  algebraically  by  simply  writing 
the  values  of  its  co-ordinates  within  parentheses,  that  of  the 
abscissa  being  always  written  first. 

Thus,  P!  (Fig.  34)  is  the  point  (3,  4),  P2  the  point 
(-  3,  4),  P3  the  point  (-  3,  -  4),  and  P4  the  point  (3,  -  4). 
In  general  the  point  whose  co-ordinates  are  x  and  y  is  the 
point  (x,  y). 


54 


SURVEYING 


This  system  of  co-ordinates  may  be  applied  to  the  determi- 
nation of  areas  in  the  following  manner : 

Suppose  the  field  to  be  ABODE  (Fig.  35).  Lay  out  the  two 
axes  so  that  the  field  shall  lie  wholly  within  the  first  quad- 
rant. Then  measure  the  co-ordinates  of  each  of  the  vertices 


M 


N      P 
FIG.  35 


R 


and  designate  them  as  follows  :  for  A,  (x^  2/1)  ;  for  B,  (x2,  2/2)  ; 
for  Cj  (xs,  2/3)  ;  for  D,  (aj4,  y^  ;  for  E,  (x6,  y5).  Evidently  each 
of  these  co-ordinates  is  positive.  Then, 

area  ABODE  =  area  LABM  -f  area  MB  OP  +  area  PCDR 

-  area  NEDR  —  area  LAEN-, 

or,  in  terms  of  the  co-ordinates, 

area  ABODE  =  ±  (2/1  +  2/2)  (x9  -  x,}  +  £  (ya  +  7/3)  (xs  -  x2) 


+  #4  (2/3    -   2/5)  +  #5(2/4  -  2/l)  ?  • 

This  method  can  be  put  in  the  form  of  a  general  rule  : 
Take  one-half  the  algebraic  sum  of  the  products  obtained 
by  multiplying  the  abscissa  of  each  vertex  by  the  difference 
between  the  ordinates  of  the  two  adjacent  vertices,  taken  in 
the  clockwise  order. 


LAND    SURVEYING 


55 


EXERCISE   V 

1.  Kequired  the  area  of  a  triangular  field  whose  sides  are 
13  chains,  14  chains,  and  15  chains. 

2.  Required  the  area  of  a  triangular  field  if  it  has  two 
angles  48°  30'  and  71°  45',  and  the  included  side  20  chains. 

3.  Required   the  area  of  a  triangular  field  whose  base  is 
12.60  chains,  and  altitude  6.40  chains. 

4.  Required  the  area  of  a  triangular  field  which  has  two  sides 
4.50  chains  and  3.70  chains,  and  the  included  angle  60°. 

5.  Required  the  area  of  a  field  in  the  form  of  a  trapezium, 
one  of  whose  diagonals  is  9  chains,  and  the  two  perpendicu- 
lars upon  this  diagonal  from  the  oppo- 
site vertices  4. 50  chains  and  3. 25  chains. 

6.  Required  the  area  of  the  field 
ABCDEF  (Fig.  36),  if 

AE=    9.25  chains,  FF*  =  6.40  chains,    B 
BE  =  13.75  chains,  DD'  =       1  chains, 
DB  =       10  chains,  CC'  =       4  chains, 

and  A  A'  —  4.75  chains.  FIG.  36 


P'PD  =  165°  40', 
P'PC  =  303°  15'. 


7.  Determine  the  area  of  the  field  ABCD  from  two  interior 
stations  P  and  P',  if  PP'  =  1.50  chains, 

PP'C=  89°  35',  PP'D  =  349°  45', 
PP'B  =  185°  30',  P'PB  =  3°  35', 
PP'A  =  309°  15',  P'PA  =  113°  45', 

8.  Required  the  area  of  the  field 
ABCDEF  (Fig.  37),  if 

AF'  =       4  chains,  FF'  =    6  chains,   A> 
EE'  =  6.50  chains,  AE'  =    9  chains, 
AD  =     14  chains,  AC'  =  10  chains, 
AB'  =  6.50  chains,  BB'  =    7  chains, 
CC'  =  6.75  chains. 


56  SURVEYING 

9.  Required  the  area  of  the  field  AGBCD  (Fig.  32,  p.  52), 
if  the  diagonal  A  C  =  5,  BB'  (the  perpendicular  from  B  to  A  C) 
=  1,  DD'  (the  perpendicular  from  D  to  A  C)  =  1.60,  EE'  = 
0.25,  FF'  =  0.25,  GG'  =  0.60,  HH'  =  0.52,  M'=  0.54,  AE'  = 
0.2,  F'F'  =  0.50,  F'G1  =  0.45,  G7T  =  0.45,  #'#'  =  0.60,  and 
K'B  =  0.40. 

10.  Required  the  area  of  the  field  AGBCD  (Fig.  33,  p.  52), 
if  AD  =  3,  AC  =  5,  AB  =  6,  angle  DAC  =  45°,  angle  BAG  = 
30°,  ylF'  =  0.75,  AF'  =  2.25,  AH  =  2.53,  4£'  =  3.15,  EE'  = 
0.60,  FF'  =  0.40,  and  GG1  =  0.75. 

11.  Determine  the  area  of  the  field  A  BCD  from  two  exterior 
stations  P  and  P',  if  PP'  =  1.50  chains, 

P'P£  -  41°  10',  P'P£>  =  104°  45',  PP'B  =  132°  15', 
P'PA  =  55°  45',  PP'D  =  66°  45',  PP'A  =  103°  0'. 
p'PC  =  77°  20',  PP'C=  95°  40', 

12.  Find  the  area  of  the  field  ABODE  (Fig.  35,  p.  54),  if 
the  co-ordinates,  in  chains,  of  the  vertices  taken  in  order  are 
(1.40,   6.75),    (4.60,    8.32),    (9.00,    9.05),    (12.15,    5.58),   and 
(5.27,  1.16). 

13.  Find  the  area  of  the  field  ABCDE  (Fig.  35,  p.  54),  by 
measuring  distances  as  follows  : 

AL  =  400  feet ;  EM  =  700  feet ;  CP  =  680  feet ; 
DR  =  380  feet ;  EN  =  200  feet;  LM  =  150  feet ; 
MN  =  250  feet;  NP  =  200  feet ;  PR  =  220  feet. 

14.  Lay  out  a  field  of  four  sides,  and  find  its  area  by  the 
method  of  triangles  and  also  by  the  method  of  rectangular 
co-ordinates. 

15.  Lay  out  a  field  of  six  sides,  and  find  its  area  by  the 
method  of  triangles  and  also  by  the  method  of  rectangular 
co-ordinates. 


LAND    SURVEYING  57 

SECTION   XIII 
GENERAL   METHOD   FOR   FARM    SURVEYS 

Definitions.  A  course  is  the  bearing  and  length  of  a 
line.  The  latitude  of  a  course  is  the  distance  between  the 
parallels  through  its  extremities,  and  is  called  a  northing  or  a 
southing,  as  the  course  is  northward  or  southward.  The 
departure  of  a  course  is  the  distance  between  the  meridians 
through  its  extremities,  and  is  called  an  easting  or  a  westing, 
as  the  course  is  eastward  or  westward.  The  meridian  dis- 
tance of  a  point  is  its  distance  from  a  meridian.  The  double 
meridian  distance  of  a  course  is  double  the  meridian  distance  of 
its  mid-point,  and  therefore  equal  to  the  sum  of  the  meridian 
distances  of  the  extremities  of  the 
course. 

Let  AB  (Fig.  38)  represent  a  line, 
whose  bearing  and  length  are  known. 
Let  MN  be  a  reference  meridian ; 
and  let  p  and  p'  be  parallels  through 
A  and  B,  and  m  and  m'  meridians 
through*  the  same  points.  Then, 
angle  mAB  represents  the  bearing 

of  line  AB.  The  latitude  of  the  course  AB  is  AE,  and  its 
departure  EB.  The  meridian  distance  of  the  point  B  is  BC 
and  of  A,  AD.  Evidently,  the  double  meridian  distance  of 
the  course  AB  is  (BC  +  AD). 

Again,  in  the  triangle  AEB, 

AE  =  AB  x  cos  EAB,     and     EB  =  AB  x  sin  EAB. 
Hence,  latitude  =  distance  x  cos  of  bearing,  and  departure  = 
distance  x  sin  of  bearing.     From  these  formulas,  the  latitude 
and  departure  of  any  course  may  be  found  by  means  of  a 
table  of  natural  sines  and  cosines.     They  may  be  found  also 


58 


SURVEYING 


from  the  Traverse  Table,  which  is  merely  the  tabulated  results 
of  the  foregoing  method  for  given  courses. 

Field  Notes.  The  field  notes  are  kept  in  a  book  provided 
for  the  purpose.  The  page  is  commonly  ruled  in  three  col- 
umns, in  the  first  of  which  is  written  the  number  of  the 
station ;  in  the  second,  the  bearing  of  the  side ;  and  in  the 
third,  the  length  of  the  side. 

FIELD  NOTES 

N 


E 


I 

N.  20°  E. 

8.66 

2 

S.  70°  E. 

5.00 

3 

S.  10°  E. 

10.00 

4 

N.  70°  W. 

10.00 

FIG.  39 


To  obtain  the  field  notes,  say  of  field 
ABCD  (Fig.  39),  place  the  compass 
at  A,  the  first  station,  and  take  the 
bearing  of  AB  (p.  12) ;  suppose  it  to 
be  N.  20°  E.  Write  the  result  in 
the  second  column  of  the  field  notes 
opposite  the  number  of  the  station. 
Measure  AB  =  8.66  chains,  and  write 
the  result  in  the  third  column  of  the 
field  notes.  Place  the  compass  at  J5, 
and,  after  testing  the  bearing  of  AB  (p.  13),  take  the  bear- 
ing of  BC,  measure  BC,  and  write  the  results  in  the  field 
notes ;  and  so  continue  until  the  bearing  and  length  of  each 
side  have  been  recorded. 

Computation  of  the  Area.  The  survey  may  begin  at  any 
corner  of  the  field  ;  but,  for  computing  the  area,  the  field  notes 
should  be  arranged  so  that  the  most  eastern  or  the  most  western 
station  shall  stand  first.  For  the  sake  of  uniformity,  we  shall 
always  begin  with  the  most  western  station  and  keep  the  field 
on  the  right  in  passing  around  it. 


LAND   SURVEYING 


59 


The  field  notes  occupy  the  first  three  of  the  eleven  columns  in  the  tablet 
below.  Columns  IV,  V,  VI,  and  VII  contain  the  latitudes  and  departures 
corresponding  to  the  sides,  taken  from  the  Traverse  Table.  The  line 
represented  by  each  number  is  indicated  immediately  above  that  number. 
Column  VIII  contains  the  meridian  distances  of  the  points  B,  C,  D,  and 
A,  taken  in  order.  Column  IX  contains  the  double  meridian  distances 


I 

II 

Ill 

IV 

V 

VI 

VII 

VIII 

IX 

X 

XI 

SIDE 

BEARING 

DlST. 

N. 

S. 

E. 

W. 

M.D. 

D.M.D. 

N.A. 

S.A. 

AB 

N.20°E. 

8.66 

AB' 
8.14 

BB' 

2.96 

BB' 

2.96 

BB' 

2.96 

2  ABB1 
24.0944 

BC 

S.  70°  E. 

'5.00 

B'C' 
1.71 

C"C 
4.70 

CC' 
7.66 

BB'  +  CC' 
10.62 

2  C'CBB' 
18.1602 

CD 

S.  10°  E. 

10.00 

CD' 
9.85 

D"D 

1.74 

DD' 
9.40 

CC'  +  DD' 
17.06 

1  D'DCC 
168.0410 

DA 

N.70°W. 

10.00 

D'A 
3.42 

DD' 

9.40 

0 

DD' 
9.40 

2  ADD 

32.1480 

.... 

33.66 

11.56 

11.56 

9.40 

9.40 

56.2424 

186.2012 

(186.2012  sq.  ch.  -  56.2424  sq.  ch.)  -=-  2  =  64.98  sq.  ch.  =  6.50  acres. 

of  the  courses.    Their  composition  is  indicated  by  the  letters  immedi- 
ately above  the  numbers.    Column  X  contains  the  products  of  the  double 
meridian  distances  by  the  northings  in  the  same  line.     The  first  number, 
24.0944  =  2.96  x  8.14  =  BB'  x  AB'  =  twice  area  of  triangle  ABB'  • 
32.1480  =  9.40  x  3.42  =  DD'  x  AD'  =  twice  area  of  triangle  ADD'. 
Column  XI  contains  the  products  of  the  double  meridian  distances  by 
the  southings  in  the  same  line.     The  first  number, 

18.1602  =  10.62  x  1.71  =  (BB'  +  CC')  x  B'C' 

—  twice  area  of  trapezoid  C'CBB'  • 
168.0410  =  17.06  x  9.85  =  (CC'  +  DD')  x  D'C' 

=  twice  area  of  trapezoid  D'DCC'. 
The  sum  of  the  north  areas  in  column  X 

=  56.2424    =  2  (ABB'  +  ADD'). 
The  sum  of  the  south  areas  in  column  XI 

=  186.2012  =  2  (C'CBB'  +  D'DCC'). 

But  (C'CBB'  +  D'DCC')  -     (ABB' +  ADD')  =    ABCD. 

Hence,     2  (C'CBB'  +  D'DCC')  -  2  (ABB'  +  ADD')  =  2  ABCD ; 
that  is,  186.2012  -  56.2424  =  129.9588  =  2  ABCD. 

Hence,  area  ABCD  =  ±  of  129.9588  sq.  ch.  =  64.98  sq.  ch.  =  6.50  A. 


60  SURVEYING 

Balancing  the  Work.  In  the  survey,  we  pass  entirely  around 
the  field  ;  hence,  we  move  just  as  far  north  as  south.  There- 
fore, the  sum  of  the  northings  should  equal  the  sum  of  the 
southings.  In  like  manner,  the  sum  of  the  eastings  should 
equal  the  sum  of  the  westings.  In  this  way  the  accuracy  of 
the  field  work  may  be  tested. 

In  the  example  on  page  59  the  sum  of  the  northings  is 
equal  to  the  sum  of  the  southings,  being  11.56  in  each  case  ;  and 
the  sum  of  the  eastings  is  equal  to  the  sum  of  the  westings, 
being  9.40  in  each  case.  Hence,  the  work  balances. 

In  actual  practice  the  work  seldom  balances.  When  it 
does  not  balance,  corrections  are  generally  applied  to  the 
latitudes  and  departures  by  the  following  rules  : 

1.  The  perimeter  of  a  field  is  to  any  one  side  as  the  total 
error  in  latitude  is  to  the  correction  required. 

2.  The  perimeter  of  a  field  is  to  any  one  side  as  the  total 
error  in  departure  is  to  the  correction  required. 

EXAMPLE.  The  perimeter  of  a  field  measured  306.62  chains 
and  one  side  72.47  chains,  with  a  total  error  of  22  links  in 
latitude  and  of  18  links  in  departure. 

Then  306.62  :  72.47  =  22  links  :  x  =  18  links  :  y. 

Whence  x  —  5  links  and  y  =  4  links. 

Hence  the  correction  in  latitude  applied  to  the  given  side  is 
0.05  chains,  and  the  correction  in  departure  is  0.04  chains. 

If  special  difficulty  was  found  in  taking  a  particular  bear- 
ing, or  in  measuring  a  particular  line,  the  corrections  should 
be  applied  to  the  corresponding  latitudes  and  departures. 

The  amount  of  error  allowable  varies  in  the  practice  of  dif- 
ferent surveyors,  and  according  to  the  nature  of  the  ground. 
An  error  of  1  link  in  8  chains  would  not  be  considered  too  great 
on  smooth,  level  ground ;  while  on  rough  ground  an  error  of 
1  link  in  3  chains  might  be  allowed.  If  the  error  is  consider- 
able, the  field  measurements  should  be  repeated. 


LAND   SURVEYING 


61 


As  another  example  let  it  be  required  to  find  the  area  of 
field  ABCDEF  from  the  following 


FIELD  NOTES 


1 

N.  73°  30'  W. 

5.00 

2 

S.  10°  30'  W. 

5.00 

3 

N.  28°  30'  W. 

7.07 

4 

N.  20°  00'  E. 

11.18 

5 

S.  43°  30'  E. 

5.00 

6 

S.  13°  30'  E. 

10.00 

SIDE 

BEARING 

DlST. 

N. 

S. 

E. 

W. 

M.D. 

D.M.D. 

N.A. 

S.A. 

AE 

N.  20°00'E. 

11  18 

10.51 

3.82 

B'B 
382 

B'B 

3  82 

1ABB' 

40.1482 

BC 
CD 

S.43°30'E. 
S  13°  30'  E 

5.00 
10  00 

3.63 
9  72 

3.44 
2  33 

... 

crc 

7.26 
D'D 
9  59 

B'B+  C'C 
11.08 
C'C  +  D'D 

16  85 

2  C'CBB' 
40.2204 
ID'DCC 
163  78°0 

DE 

N.  73°30'W 

5.00 

1  42 

4.79 

E'E 
4.79 

D'D  +  E'E 
14.38 

2  D'DEE' 
20.4196 

EF 
FA 

S.1G°30'W. 
N  28°30'W 

5.00 
707 

6  21 

4.79 

... 

4.80 
1.42 

3  37 

F'F 
3.37 

0  00 

E'E  +  F'F 
8.16 
F'F 
3  37 

2AFF' 
20  9277 

2  F'FEE' 
39.0864 

43.25 

18.14 

18.14 

9.59 

9.58 
9.59 

81.4955 

243.0888 

43.25  :  5  =  0.01  :  x.         Area  =  8.08  acres. 

The  first  station  in  the  field  notes  is  Z),  but  we  rearrange  the  numbers 
in  the  tablet  so  that  A  stands  first.  The  northings  and  southings  balance, 
but  the  eastings  exceed  the  westings  by  1  link.  We  apply  the  correction 
to  the  westing  4.79  (the  distance  DE  being  in  doubt),  making  it  4.80,  and 
write  the  correction.  In  practice,  the  corrected  numbers  are  written  in 
red  ink,  and  often  all  the  latitudes  and  departures  are  rewritten  in  four 
additional  columns,  headed,  respectively,  N',  S',  E',  W'. 

Supplying  Omissions.  If  for  any  reason  the  bearing  and  the 
length  of  any  side  do  not  appear  in  the  field  notes,  the  lati- 
tude and  departure  of  this  side  may  be  found  in  the  following 
manner : 


62 


SURVEYING 


Find  the  latitudes  and  departures  of  the  other  sides  as 
usual.  The  difference  between  the  northings  and  southings 
gives  the  northing  or  southing  of  the  unknown  side,  and  the 
difference  between  the  eastings  and  westings  gives  the  easting 
or  westing  of  the  unknown  side. 

If  the  length  and  the  bearing  of  the  unknown  side  are  de- 
sired, they  may  be  found  by  solving  the  right  triangle,  whose 
sides  are  the  latitude  and  departure  found  by  the  method  just 
explained,  and  whose  hypotenuse  is  the  length  required. 

Obstructions.  If  the  end  of  a  line  is  not  visible  from 
its  beginning,  or  if  the  line  is  inaccessible,  its  length  and 
bearing  may  be  found  as  follows : 

By  means  of  a  random  line  (p.  8). 

When  it  is  impossible  to  run  a  random  line,  which  is  fre- 
quently the  case  on  account  of  the  extent  of  the  obstruction, 
the  following  method  may  be  used : 


N 


Let  AB  (Fig.  40)  represent  an  inaccessible  line 
whose  extremities  A  and  B  only  are  known,  and  B 
invisible  from  A. 

Set  flagstaffs  at  convenient  points,  C  and  D. 
Find  the  bearings  and  lengths  of  AC,  CD,  and  DB, 
and  then  proceed  to  find  the  latitude  and  departure 
of  AB. 


FIG.  40 


EXAMPLE. 
lowing  notes  (see  Fig.  40) : 


Suppose  that  we  have  the  f  ol- 


SIDE 

BEARING 

DlST. 

N. 

S. 

E. 

w. 

AC 

S.  45°  E. 

3.00 

2.12 

2.12 

CD 

E. 

3.50 

3.50 

DB 

N.  30°  E. 

4.83 

4.18 

2.42 

4.18 

2.12 

8.04 

0 

LAND    SURVEYING 


63 


The  northing  of  AB  is  AE  =  2.06,  and  the  easting,  EB  =  8.04.  These 
numbers  may  be  entered  in  the  tablet  in  the  columns  N.  and  E.,  opposite 
the  side  AB. 

If  the  bearing  and  length  of  AB  are  required, 

tan  BAE  =  ?^_  =  ^  =  3.903. 
AE      2.06 

Hence,  the  angle  BAE  =  75°  38'. 

Also, 


AB  =  ^AE   +  ~BE*  =  V8.042  +  2.06*  =  8.30. 
Therefore,  the  bearing  and  length  of  AB  are  N.  75°  38'  E.  and  8.30. 

To  make  a  Plot.     A   plot  or  map  may  be  drawn  to  any 
desired  scale.     If  a  line  1  inch  in  length  in  the  plot  represents 
a  line  1  chain  in  length,  the  plot  is 
said  to  be  drawn  to  a  scale  of  1  chain 
to  an  inch.    In  this  case  (Fig.  41)  the 
plot  is  drawn  to  a  scale  of  8  chains 
to  an  inch. 

Draw  the  line  NAS  to  represent 
the  meridian,  and  lay  off  the  first 
northing  ^£'  =  8.14.  Through  B 
draw  an  indefinite  line  perpendicular 
to  NS  and  lay  off  B'B,  the  first  easting, 
=  2.96.  Draw  AB  •  then  the  line  AB 
represents  the  first  side  of  the  field. 
Through  B  draw  BC"  perpendicular 
to  BB',  and  make  BC"  =  1.71,  the  first 
southing.  Through  C"  draw  C"C 
perpendicular  to  BC",  and  equal  to 
4.70,  the  second  easting.  Draw  BC. 
The  line  BC  represents  the-  second 
side  of  the  field.  Proceed  in  like 
manner  until  the  field  is  completely 
represented.  The  extremity  of  the 
last  line  F'A,  measured  from  F',  should  fall  at  A. 
test  of  the  accuracy  of  the  plot. 


F    F 


FIG.  41 


This  is  a 


64 


SURVEYING 


By  drawing  AC,  AE,  and  EC,  the  hexagonal  figure 
ABCDEFA  is  divided  into  triangles,  the  bases  and  altitudes 
of  which  may  be  measured  and  the  area  computed  approxi- 
mately. 

Another  method  is  as  follows  : 
Draw  MN  (Fig.  42)  to  represent  a 
meridian.  Let  the  point  A  in  this 
line  be  taken  as  the  first  station  in 
the  rearranged  field  notes  of  page 
61.  With  the  circular  protractor 
mark  off  each  of  the  bearings  as  I,  c, 
dj  e,  fj  and  a.  Draw  AE  to  scale 
through  b.  With  triangle  and  ruler 
(p.  48)  or  with  parallel  ruler  draw  to 
scale  EC  parallel  to  Ac  ;  and  so  on. 

After  some  practice,  still  other 
methods  will  be  suggested,  but  the 
methods  given  are  among  the  best. 


EXERCISE   VI 

Find  the  areas  of  the  following  and  make  a  plot  of  each. 
In  3  and  7,  detours  were  made  on  account  of  obstructions 
(p.  62).     The  notes  of  the  detours  are  written  in  braces. 

123 


STA. 

BEARINGS 

DlST. 

1 

S.  75°  E. 

6.00 

2 

S.  15°  E. 

4.00 

3 

S.  75°  W. 

6.93 

4 

N.  45°  E. 

5.00 

5 

N.  45°  W. 

5.19* 

STA. 

BEARINGS 

DlST. 

1 

N.  45°  E. 

10.00 

2 

S.  75°  E. 

11.55 

3 

S.  15°  W. 

18.21 

4 

N.  45°  W. 

19.11 

STA. 

BEARINGS 

DlST. 

1 

S.    2°15/E. 

9.68 

J 

N.  51°45'W. 
S.  85°00'W. 

2.39 
6.47 

[ 

S.  55°10/W. 

1.62 

3 

N.    3°45'E. 

6.39 

4 

S.  66°45'E. 

1.70 

5 

N.  15°00'E. 

4.98 

6 

S.  82°45'E. 

6.03 

LAND   SURVEYING 


STA. 

BEARINGS 

DlST. 

1 

N.    5°30'W. 

6.08 

2 

S.  82°30/W. 

6.51 

3 

S.    3°00'E. 

5.33 

4 

E. 

6.72 

STA. 

BEARINGS 

DlST. 

1 

N.    6°15'W. 

6.31 

2 

S.  81°50'W. 

4.06 

3 

S.    5°00'E. 

5.86 

4 

N.  88°30'E. 

4.12 

STA. 

BEARINGS 

DlST. 

1 

N.  20°00'E. 

4.62i 

2 

N.  73°00'E. 

4.16| 

3 

S.  45°15'E. 

6.18i 

4 

S.  38°30'W. 

8.00 

5 

Wanting 

Wanting 

STA. 

BEARINGS 

DlST. 

,  r 

S.  81°20'W. 

4.28 

M 

N.  76°30'W. 

2.67 

2 

N.    5°00'E. 

8.68 

3 

S.  87°30/E. 

5.54 

r 

S.    7°00'E. 

1.79 

4 

S.  27°00'E. 

1.94 

| 

S.  10°30'E. 

5.35 

I 

N.  76°45/W. 

1.70 

STA. 

BEARINGS 

DlST. 

1 

N.  89°45'E. 

4.94 

2 

S.    7°00/W. 

2.30 

3 

S.  28°00/E. 

1.52 

4 

S.    0°45'E. 

2.57 

5 

N.  84°45/W. 

5.11 

6 

N.    2°30'W. 

5.79 

9.  An  Ohio  farm  is  bounded  and  described  as  follows : 
Beginning  at  the  southwest  corner  of  lot  No.  13,  thence  N.  1^° 
E.  132  rods  and  23  links  to  a  stake  in  the  west  boundary 
line  of  said  lot ;  thence  S.  89°  E.  32  rods  and  15T4o  links  to 
a  stake  5  thence  N.  1  J°  E.  29  rods  and  15  links  to  a  stake 
in  the  north  boundary  line  of  said  lot ;  thence  S.  89°  E.  61 
rods  and  18/o  links  to  a  stake;  thence  S.  32£°  W.  54  rods 
to  a  stake;  thence  S.  35£°  E.  22  rods  and  4  links  to  a 
stake ;  thence  S.  48°  E.  33  rods  and  2  links  to  a  stake ; 
thence  S.  7£°  W.  76  rods  and  20  links  to  a  stake  in  the  south 
boundary  line  of  said  lot ;  thence  N.  89°  W.  96  rods  and  10 
links  to  the  place  of  beginning.  Containing  85.87  acres,  more 
or  less. 

Verify  the  area  given  and  plot  the  farm. 


66  SURVEYING 

Modification  of  the  Latitude  and  Departure  Method.  The  area 
of  a  field  may  be  found  by  a  modification  of  the  latitude  and 
departure  method,  if  its  sides  and  interior  angles  are  known. 

Let  A,  B,  Cj  D  represent  the  interior  angles  of  the  field 
ABCD  (Fig.  43).  Let  the  side  AB  determine  the  direction  of 
reference.  The  bearing  of  AB,  with  reference  to  AB,  is  0°. 
The  bearing  of  BC,  with  reference  to  AB,  is  the  angle 
b  ==  180°  -  B.  The  bearing  of  CD,  with  reference  to  AB,  is 
the  angle  c  =  C  —  b.  The  bearing  of  DA,  with  reference 

to  AB,  is  the  angle  d  =  A. 

The  area  may  now  be  com- 
puted by  the  latitude  and  depar- 
ture method,  regarding  AB  as  the 
meridian. 

In  practice,  the  exterior  angles, 
when  acute,  are  usually  measured. 
As  the  interior  angles  may  be 
measured  with  considerable  accu- 
racy by  the  transit,  the  latitudes 

and  departures  should  be  obtained  by  using  a  table  of  natural 
sines  and  cosines. 

EXERCISE    VII 

1.  Find  the  area  of  the  field  ABCD,  in  which  the  angle 
A  =  120°,    B  =  60°,    C  =  150°,    and    D  =  30° ;    and    the   side 
AB  =  4  chains,  BC  =  4  chains,  CD  =  6.928  chains,  and  DA  = 
8  chains. 

Keep  three  decimal  places,  and  use  the  Traverse  Table. 

2.  Find  the  area  of  the  farm  ABCDE,  in  which  the  angle 
A  =  106°  19',  B  =  99°  40',  C  =  120°  20',  D  =  86°  8',  and  E  = 
127°  33';   and  the  side  AB  =  79.86  rods,  BC  =  121.13  rods, 
CD  =  90  rods,  DE  =  100.65  rods,  and  EA  =  100  rods. 

Use  the  table  of  natural  sines  and  cosines,  keeping  two  decimal  places 
in  the  results. 


LAND   SURVEYING  67 

General  Remarks  on  determining  Areas.  Operations  depend- 
ing upon  the  reading  of  the  magnetic  needle  must  lack 
accuracy.  Hence,  when  great  accuracy  is  required  (which  is 
seldom  the  case  in  land  surveying)  the  method  of  pp.  58-61 
cannot  be  employed. 

The  best  results  are  obtained  by  the  methods  explained  on 
pp.  51-54  and  66,  the  horizontal  angles  being  measured  with 
the  transit,  and  great  care  exercised  in  measuring  the  lines. 


SECTION   XIV 
LOCATION    SURVEYS 

Definition.  In  surveying  proper  we  measure  lines  and 
angles  as  we  find  them,  while  in  location  surveys  we  mark  them 
out  on -the  ground  where  they  are  required  to  be  in  order  to 
inclose  a  given  area,  or  conform  to  a  specified  shape,  or  meet 
some  other  given  condition.  Laying  out,  parting  off,  and 
dividing  up  land  are  included  in  this  class  of  surveys.  The 
surveyor  must,  for  the  most  part,  depend  on  his  general  knowl- 
edge of  Geometry  and  Trigonometry,  and  his  own  ingenuity, 
for  the  solutions  of  problems  that  arise  in  location  surveys. 

Illustrative   Problems.      PROBLEM    1.      To    divide    a   trian- 
gular field  into  two  parts  having  a  given 
ratio,  by  a  line  through  a  given  vertex. 

Let  ABC  (Fig.  44)  be  the  triangle,  and  A  the 
given  vertex. 

T>  T\ 

Divide  BC  at  Z>,  so  that equals  the  given 

X/(_/ 

ratio,  and  draw  AD.    ABD  and  ADC  are  the  parts     fc________. _______ 

required;  for  D 

ABD  :  ADC  =  BD  :  DC.  ™-  ** 

PROBLEM  2.  To  cut  off  from  a  triangular  field  a  given 
area,  by  a  line  parallel  to  the  base. 


68 


SURVEYING 


Let  ABC  (Fig.  45)  be  the  triangle,  and 
let  DE  be  the  division  line  required. 

Then  ABC  :  A  DE  =  AS?  :  ZZ>2. 

•"•  VABC  :  VADE  =  AB  •.  AD.  . 


.-.  AD  = 


ADE 

ABC' 


PROBLEM  3.     To  cut  off  from  a 
FIG.  45  triangular  field  a  given  fraction  of 

the  field,  by  a  line  from  a  given  point  in  a  side. 

Let  ABC  (Fig.  46)  be  the  triangle,  and  P  the  point  from  which  the 
line  PD  is  to  be  located  so  as  to  cut  off,  say,  one-third  the  area  of  the 
triangle. 

AD  =  AB  x  AC  +  SAP. 

For  ABC:APD  =  AB  x  AC  :  AP  x  AD  =  3:1. 


FIG.  4G 

PROBLEM  4.  To  divide  any  field  into  two  parts  having  a 
given  ratio,  by  a  line  through  a  given  point  in  the  perimeter. 

Let  ABCDE  (Fig.  47)  represent  the  field,  P  the  given  point,  and  PQ 
the  required  division  line. 

The  areas  of  the  whole  field  and  of  the  required  parts  having  been 
determined,  run  the  line  PD  from  P  to  a  corner  D,  dividing  the  field, 
approximately,  as  required.  Determine  the  area  PBCD. 

The  triangle  PDQ  represents  the  part  which  must  be  added  to  PBCD 
to  make  the  required  division. 


LAND   SURVEYING 


69 


Hence, 


NOTE. 


Area  PDQ  =  i  x  PD  x  DQ  x  sin  PDQ. 

2  x 


PD  x  sin  PDQ 


2  x  area  PDQ 


This  perpendicular  from 


perpendicular  from  P  on  DE 
P  on  DE  may  be  run  and  measured  directly. 

PROBLEM  5.     To  divide  a  field  into  a  given  number  of  parts, 
so  that  access  to  a  pond  of  water  is 
given  to  each. 

Let  ABODE  (Fig.  48)  represent  the  field, 
and  P  the  pond.  Let  it  be  required  to  divide 
the  field  into  four  parts.  Find  the  area  of 
the  field  and  of  each  part. 

Let  AP  be  one  division  line.  Run  PE, 
and  find  the  area  APE.  Take  the  differ- 
ence between  APE  and  the  area  of  one 
of  the  required  parts ;  this  gives  the  area  of 
the  triangle  PQ#,  from  which  QE  may  be 
found,  as  in  Problem  4.  Draw  PQ;  PAQ  is 
one  of  the  required  parts.  In  like  manner, 
PQR  and  PAS  are  determined;  whence, 
PSR  must  be  the  fourth  part  required. 


E 


EXERCISE   Vm 

1.  From  the  square  A  BCD,  containing  6  acres  1  rood  24 
perches,  part  off  3  acres  by  a  line  EF  parallel  to  AB. 

2.  From  the  rectangle  ABCD,  containing  8  acres  1   rood 
24  perches,  part  off  2  acres  1  rood  32  perches  by  a  line  EF 
parallel  to  AD  which  is  equal  to  7  chains.     Then,  from  the 
remainder  of  the  rectangle,  part  off  2  acres  3  roods  25  perches, 
by  a  line  GH  parallel  to  EB. 

3.  Part  off  6  acres  3  roods  12  perches  from  a  rectangle  ABCD, 
containing  15  acres,  by  a  line  EF  parallel  to  AB ;   AD  being 
10  chains. 


70  SURVEYING 

4.  From  a  square  ABCD,  whose  side  is  9  chains,  part  off  a 
triangle  which  shall  contain  2  acres  1  rood  36  perches,  by  a 
line  BE  drawn  from  B  to  the  side  AD. 

5.  From  ABCD,  representing  the  rectangle,  whose  length  is 
12.65  chains,  and  breadth  7.58   chains,  part  off  a  trapezoid 
which  shall  contain  7  acres  3  roods  24  perches,  by  a  line  BE 
drawn  from  B  to  the  side  DC. 

6.  In  the  triangle  ABC,  AB  =  12  chains,  AC  =  10  chains, 
and  BC  =  8  chains ;   part  off  a  trapezoid  of  1  acre  2  roods  16 
perches,  by  the  line  DE  parallel  to  AB. 

1.  In  the  triangle  ABC,  AB  =  26  chains,  AC  =  20  chains, 
and  BC  =  16  chains ;  part  off  a  trapezoid  of  6  acres  1  rood  24 
perches,  by  the  line  DE  parallel  to  AB. 

8.  It  is  required  to  divide  the  triangular  field  ABC  among 
three  persons  whose  claims  are  as  the  numbers  2,  3,  and  5,  so 
that  they  may  all  have  the  use  of  a  watering  place  at  C;   AB 
=  10  chains,  AC  =  6.85  chains,  and  CB  =  6.10  chains. 

9.  Divide  the  five-sided  field  ABCHE  among  three  persons, 
X,  Y,  and  Z,  in  proportion  to  their  claims,  X  paying  $500,  Y 
paying  $750,  and  Z  paying  $1000,  so  that  each  may  have  the 
use  of  an  interior  pond  at  P,  the  quality  of  the  land  being 
equal  throughout.     Given  AB  =  8.64  chains,  BC  =  8.27  chains, 
CH  =  8.06  chains,  HE  =  6.82  chains,  and  EA  =  9.90  chains. 
The  perpendicular  PD  upon  AB  =  5.60  chains,  PD1-  upon  BC 
=  6.08  chains,  PD"  upon   CH  =  4.80  chains,  PD'"  upon  HE 
=  5.44  chains,  and  PD""  upon  EA  =  5.40  chains.     Assume 
PH  as  the  divisional  fence  between  the  shares  of  X  and  Z,  it 
is  required  to  determine  the  position  of  the  fences  PM  and  PN 
between  the  shares  of  X  and  Y  and  between  the  shares  of  Y 
and  Z. 

10.  Divide  the  triangular  field  ABC,  whose  sides  AB,  AC, 
and  BC  are  15,  12,  and  10  chains,  respectively,  into  three  equal 


LAND   SURVEYING  71 

parts,  by  fences  EG  and  DF  parallel  to  BC,  without  finding 
the  area  of  the  field. 

11.  Divide  the  triangular  field  ABC,  whose  sides  AB,  BC, 
and  AC  are  22,  17,  and  15  chains,  respectively,  among  three 
persons,  A,  B,  and  C,  by  fences  parallel  to  the  base  AB,  so  that 
A  may  have  3  acres  above  the  line  AB,  B  4  acres  above  A's 
share,  and  C  the  remainder. 


SECTION   XV 
LAYING    OUT   THE   PUBLIC   LANDS 

Reference  Lines.  The  public  lands  north  of  the  Ohio  Eiver 
and  west  of  the  Mississippi  are  generally  laid  out  in  accord- 
ance with  what  is  known  as  the  rectangular  system  of 
surveying.  First,  an  initial  point  is  selected  with  great  care, 
and  then  astronomically  established.  Through  this  point  a 
principal  meridian,  or  true  north  and  south  line,  is  run  by 
means  of  the  solar  compass,  or  the  transit  with  observations 
on  Polaris  ;  and  also  an  east  and  west  line,  called  a  base  line. 
Crossing  the  principal  meridian  at  intervals  of  24  miles,  both 
north  and  south  of  the  initial  point,  are  run  other  east 
and  west  lines,  called  standard  parallels,  or  correction  lines. 
Northward  from  the  base  line  and  from  each  of  the  standard 
parallels,  at  intervals  of  24  miles,  both  ways  from  the  princi- 
pal meridian,  are  run  true  north  and  south  lines,  called  guide 
meridians.  Thus,  the  land  is  divided  into  blocks  approxi- 
mately 24  miles  square.  Six  principal  meridians  have  been 
established,  in  addition  to  which  and  connected  with  wfyich 
there  are  twenty  or  more  independent  meridians  in  the 
western  states  and  territories. 

Division  from  Reference  Lines;  Townships.  Within  each 
block  parallels  to  the  base  line,  or  to  a  standard  parallel,  are 
run  at  intervals  of  6  miles.  These  are  called  township  lines. 


72 


SURVEYING 


At  the  same  intervals  are  also  run  north  and  south  lines, 
called  range  lines.  Thus,  the  tract  would  be  divided  into 
townships  exactly  6  miles  square  if  it  were  not  for  the  con- 
vergence of  the  meridians  on  account  of  the  curvature  of  the 
earth.  An  east  and  west  series  of  townships  is  called  a  tier, 
and  a  north  and  south  series  is  called  a  range.  A  township 
is  designated  by  giving  the  number  of  the  tier  north  or  south 
of  the  base  line  and  the  number  of  the  range  east  or  west  of 


TF- 


D- 

M' 

1 

O 

i\k  A    i\  M 

T 

t" 

*o" 

B 

t1 

p' 

. 

£ 

A 

p 

r 

r' 

r'' 

\ 

G 

u 

G 

D 

C 

-E 


FIG.  49 

the  principal  meridian.  Thus,  T.  3  N.,  R.  2  W.,  read  town- 
ship three  north,  range  two  west,  means  that  the  township  is 
in  the  third  tier  north  of  the  base  line,  and  in  the  second  tier 
west  of  the  principal  meridian. 

Let  NS  (Fig.  49)  represent  a  principal  meridian  ;   WE  a  base 
line;   DL  and  D'L'  standard  parallels;    GM  and  G'M1  guide 


LAND   SURVEYING 


73 


meridians ;  rl,  r'l',  .  .  .  ,  range  lines  ;  tpt  t'p',  .  .  .  ,  township 
lines.  If  Or  is  taken  as  6  miles,  then  O'l  will  be  less  than 
6  miles.  O'k  being  equal  to  6  miles  and  O'l  being  less, 
it  will  be  observed  that  there  will  be  offsets  on  the  base  line 
and  on  standard  parallels  at  intervals  of  6  miles. 

Township  A  would  be  designated  thus  :  T.  2  X.,  R.  3  E. 
How  would  townships  B  and  C  be  designated? 

Subdivision  of  Townships.  The  townships  are  divided  into 
sections  approximately  1  mile  square,  and  the  sections  are 
divided  into  quarter  sections.  The 
township,  section,  and  quarter-section 
corners  are  permanently  marked.  The 
sections  are  numbered,  beginning  at 
the  northeast  corner,  as  in  Fig.  50, 
which  represents  a  township  divided 
into  sections.  The  quarter  sections  are 
designated,  according  to  their  position, 
as  N.E.,  N.W.,  S.E.,  and  S.W.  Section 
lines  are  surveyed  in  such  an  order  as 

to  throw  the  errors  on  the  northwest  quarter  sections,  which 
are  carefully  measured  and  their  areas  calculated. 

Meander  Lines.  If  in  running  a  line  a  navigable  stream 
or  a  lake  more  than  1  mile  in  length  is  encountered,  it  is 
meandered  by  marking  the  intersection  of  the  line  with  the 
bank  and  running  lines  from  this  point  along  the  bank  to 
prominent  points  which  are  marked,  and  the  lengths  and 
bearings  of  the  connecting  lines  recorded. 

Manual.  For  detail  of  methods,  see  the  "Manual  of  Sur- 
veying Instructions,"  issued  by  .the  Commissioner  of  the 
General  Land  Office,  at  Washington,  D.C.,  for  the  use  of 
Surveyors-General. 


G 

5 

4 

3 

2 

1 

7 

8 

9 

10 

It 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

FIG.  so 


CHAPTER   IV 


TRIANGULATION 

SECTION  XVI 

DEFINITIONS 

The  third  method  of  surveying  explained  on  pa^;  51  is 
an  example  of  triangulation  on  a  small  scale.  The  simple 
principle  there  involved  is  elaborately  worked  out  in  hydro- 
graphic  or  topographic- surveys,  or  in  the  measurement  of 
terrestrial  arcs,  as  in  the  "  Transcontinental  Triangulation 
and  American  Arc  of  the  Parallel,"  recently  completed  by 
the  United  States  Coast  and  Geodetic  Survey. 

Let  F  (Fig.  51)  represent  a  point  whose  position  with 
reference  to  the  base  line  AB  is  required.  Connect  AB  with 
F  by  the  series  of  triangles  Q 

ABC,  A  CD,  ADE,  and  DEF, 
so  that  a  signal  at  C  is  visible 
from  A  and  B,  a  signal  at  D 
visible  from  A  and  C,  a  signal 
at  E  visible  from  A  and  Z>, 
and  a  signal  at  F  visible  from 
D  and  E.  In  the  triangle  AB  C, 
the  side  AB  is  known,  and  the  angles  at  A  and  B  may  be 
measured;  hence,  AC  may  be  computed.  In  the  triangle 
ACD,  AC  is  known,  and  the  angles  at  A  and  C  may  be 
measured  ;  hence,  AD  may  be  computed.  In  like  manner, 
DE  and  EF  or  DF  may  be  determined.  DF,  or  some  suitable 
line  connected  with  DF,  may  be  measured,  and  this  result 

74 


FIG.  51 


TRIANGULATION  75 

compared  with  the  computed  value  to  test  the  accuracy  of 
the  field  measurement.  This  net  or  chain  of  triangles  enables 
us  to  determine  the  relative  position  of  all  the  points  with 
respect  to  each  other.  If  the  point  A  is,  furthermore, 
astronomically  located,  and  the  azimuth  of  line  AB  is  known, 
then  we  have  sufficient  data  also  to  determine  the  absolute 
geographical  position  of  each  of  the  points. 

Classification.  Three  orders  of  triangulation  are  recognized, 
viz. :  primary,  in  which  the  sides  are  from  20  to  190  miles  in 
length;  secondary,  in  which  the  sides  are  from  5  to  40  miles 
in  length,  and  which  connect  the  primary  with  the  tertiary ; 
tertiary,  in  which  the  sides  are  seldom  over  5  miles  in  length, 
and  which  bring  the  survey  down  to  such  dimensions  as  to 
admit  of  the  minor  details  being  filled  in  by  the  compass  and 
plane  table. 

Measurement  of  Base  Lines.  Base  lines  should  be  measured 
with  a  degree  of  accuracy  corresponding  to  their  importance. 
Suitable  ground  must  be  selected  and  cleared  of  all  obstruc- 
tions. Each  extremity  of  the  line  may  be  marked  by  cross 
lines  on  the  head  of  a  copper  tack  driven  into  a  stub  which 
is  sunk  to  the  surface  of  the  ground.  Poles  are  set  up  in 
line  about  half  a  mile  apart,  the  alignment  being  controlled 
by  a  transit  or  theodolite  placed  over  one  end  of  the  line. 
The  preliminary  measurement  may  be  made  with  an  iron 
wire  about  one-eighth  of  an  inch  in  diameter  and  60  meters 
in  length,  or  with  a  steel  chain  of  the  same  length. 

The  final  measurement  is  made  with  the  tape  line,  or  with 
bars  6  meters  long,  which  are  supported  upon  trestles  when  in 
use.  These  bars  are  placed  end  to  end,  and  brought  to  a 
horizontal  position,  if  this  can  be  quickly  accomplished ;  if 
not,  the  angle  of  inclination  is  taken  by  a  sector,  or  a  vertical 
offset  is  measured  with  the  aid  of  a  transit,  so  that  the  exact 
horizontal  distance  can  be  computed.  A  thermometer  is 
attached  to  each  bar,  so  that  the  temperature  of  the  bar  may 


76  SURVEYING 

be  noted  and  a  correction  for  temperature  applied.  Some- 
times the  bars  are  laid  in  melting  ice,  in  which  case  accuracy 
to  at  least  one  five-millionth  part  of  the  length  measured  is 
attainable. 

Measurement  of  Angles.  Angles  are  measured  by  means  of 
the  transit  with  much  greater  accuracy  than  with  the  com- 
pass, since  the  reading  of  the  plates  of  the  transit  is  taken  to 
minutes,  and  by  means  of  microscopes  to  seconds,  while  the 
reading  of  the  needle  of  the  compass  is  to  quarter  or  half- 
quarter  degrees. 

In  order  to  eliminate  errors  of  observation  and  of  adjust- 
ment, and  errors  arising  from  imperfect  graduation  of  the 
circles,  a  large  number  of  readings  is  made  and  their  mean 
taken.  Two  methods  are  in  use,  viz.,  repetition  and  series. 

The  method  of  repetition  consists  essentially  in  taking  as 
many  readings  of  an  angle  as  is  desired,  the  reading  in  each 
case  after  the  first  being  from  the  index  of  the  next  preceding 
reading,  and  then  taking  the  mean. 

The  method  of  series  is  the  one  generally  used  when  several 
angles  about  the  same  point  are  to  be  measured.  It  con- 
sists essentially  in  taking  the  readings  successively  on  each 
station,  then  reversing  the  telescope  and  repeating  the  obser- 
vations in  the  reverse  order,  which  completes  a  series.  This 
process  is  repeated  a  number  of  times,  each  series  beginning 
with  a  different  index.  Then  the  mean  of  the  different  series 
is  found. 

On  account  of  the  curvature  of  the  earth,  the  sum  of  the 
three  angles  of  a  triangle  upon  its  surface  exceeds  180°. 
This  spherical  excess,  as  it  is  called,  becomes  appreciable  only 
when  the  sides  of  the  triangle  are  about  5  miles  in  length. 
To  determine  the  angles  of  the  rectilinear  triangle  having  the 
same  vertices,  one-third  of  the  spherical  excess  is  generally 
deducted  from  each  spherical  angle. 


CHAPTER   V 

LEVELING 
SECTION   XVII 

DEFINITIONS 

A  level  surface  is  a  surface  parallel  with  the  surface  of 
still  water,  and  is,  therefore,  slightly  curved  owing  to  the 
spheroidal  shape  of  the  earth.  A  level  line  is  a  line  in  a 
level  surface.  The  line  of  apparent  level  of  a  place  is  a 
tangent  to  the  level  line  at  that  place.  Hence,  the  line  of 
apparent  level  is  perpendicular  to  the  plumb  line. 

Leveling  is  the  process  of  finding  the  difference  of  level  of 
two  places,  or  the  distance  of  one  place  above  or  below  a  level 
line  through  another  place. 

Corrections  for  Curvature  and  Refraction.  In  ordinary  leveling 
no  distinction  is  made  between  true  aad  apparent  levels.  In 
precise  leveling  the  difference  between  the  two  is  measured, 
i.e.,  correction  is  made  for  curvature  of  the  earth.  There  is 
sometimes  also  a  correction  made  for  refraction  of  light. 

Let  t  (Fig.  52)  represent  the  line  of  apparent  level  of  the 
place  P,  a  the  level  line,  d  the  diameter  t  p 

of  the  earth  ;  then  c  represents  the  cor- 
rection for  curvature.  To  compute  the 
correction  for  curvature  : 

t2  =  c(c  +  d).     (Geometry,  §  381.) 


2 


Therefore,    c  =  -  ;  =  —  ?    approxi- 
c  +  d       d 

mately,  since  c  is  very  small  compared 

with  d,  and  t  =  a,  very  nearly.  FJG.  52 

77 


78  SURVEYING 

Since  d  is  constant  (=  7920  miles,  nearly),  the  correction 
for  curvature  varies  as  the  square  of  the  distance. 

EXAMPLE.     What  is  the  correction  for  curvature  for  1  mile  ? 
By  substituting  in  the  formula  deduced  above, 

miles  =  8  inches,  nearly. 


7920 

Hence,  the  correction  for  curvature  for  any  distance  may 
be  found  in  inches,  approximately,  by  multiplying  8  by  the 
square  of  the  distance  expressed  in  miles. 

If  correction  for  refraction  is  also  made,  it  is  customary  to 
diminish  the  above  by  about  one-sixth  of  itself;  or,  c  =  f  of  8  a2. 

SECTION   XVIII 
DIFFERENTIAL   LEVELING 

Single  Setting  of  Instrument.  To  find  the  difference  of 
level  between  two  places  when  both  are  visible  from  some 
intermediate  point,  and  the  difference  of  level  does  not 
exceed  13  feet,  only  one  setting  of  the  level  will  usually  be 
necessary. 

Let  A  and  B  (Fig.  53)  represent  the  two  places.  Set  the 
Y  level  at  a  station  equally  distant,  or  nearly  so,  from  A  and 


FIG.  53 


B,  but  not  necessarily  on  the  line  AB.  After  leveling  the 
instrument,  bring  the  telescope  to  bear  upon  the  rod  (p.  38), 
and  by  signal  direct  the  rodman  to  move  the  target  until  its 
horizontal  line  is  in  the  line  of  apparent  level  of  the  telescope. 


LEVELING 


79 


Let  the  rodman  now  record  the  height  A  A'  of  the  target.  In 
like  manner  find  BB'.  The  difference  between  A  A'  and  BB' 
is  the  difference  of  level  required.  If  the  instrument  is 
equally  distant  from  A  and  B,  or  nearly  so,  the  curvature  and 
the  refraction  on  the  two  sides  of  the  instrument  balance,  and 
no  correction  for  curvature  or  refraction  is  necessary. 

Several  Settings  of  Instrument.  When  both  places  are  not 
visible  from  the  same  place,  or  when  the  difference  of  level 
between  them  is  considerable,  two  or  more  settings  of  the 
level  may  be  necessary. 

Let  A  and  D  (Fig.  54)  represent  the  two  places.  Place  the 
level  midway  between  A  and  some  intermediate  station  B. 


FIG.  54 

Find  A  A'  and  BB',  as  in  the  preceding  case,  and  record  the 
former  as  a  backsight  and  the  latter  as  a  foresight.  Select 
another  intermediate  station  C,  and  in  like  manner  find  the 
backsight  BB"  and  the  foresight  CC";  and  so  continue  until 
the  place  D  is  reached. 

The  difference  between  the  sum  of  the  foresights  and  the  sum 
of  the  backsights  ivill  be  the  difference  of  level  required. 

Since,  BB'  +  CC'  +  DD'  -  (A A'  +  BB"  +  CC") 
=  BB'  -  BB"  +  CC'  -  CC"  +  DD'  -  A  A' 
=  B'B"  +  C'C"  +  D'D  -  AA'  =  A'A"  -  AA'  =  AA". 


80  SURVEYING 

SECTION   XIX 
PROFILE   LEVELING 

Definitions.  The  intersection  of  a  vertical  plane  with  the 
surface  of  the  earth  is  called  a  section,  or  profile.  The  term 
"  profile/'  however,  usually  designates  the  plot,  or  representa- 
tion of  the  section  on  paper. 

Profile  leveling  is  leveling  to  obtain  the  data  necessary  for 
making  a  profile  or  plot  of  any  required  section. 

A  profile  is  made  for  the  purpose  of  exhibiting  in  a  single 
view  the  inequalities  of  the  surface  of  the  ground  for  great 
distances  along  the  line  of  some  proposed  improvement,  such 
as  a  railroad,  canal,  or  ditch,  thus  facilitating  the  establishment 
of  the  proper  grades. 

The  data  necessary  for  making  a  profile  of  any  required 
section  are  the  heights  of  its  different  points  above  some 
assumed  horizontal  plane,  called  the  datum  plane,  together 
with  their  horizontal  distances  apart  or  their  distances  from 
the  beginning  of  the  section. 

The  position  of  the  datum  plane  is  fixed  with  reference  to 
some  permanent  object  near  the  beginning  of  the  section, 
called  a  bench  mark,  and  in  order  to  avoid  negative  heights 
is  assumed  at  such  a  distance  below  this  mark  that  all  the 
points'  of  the  section  shall  be  above  it. 

The  heights  of  the  different  points  of  the  section  above 
the  datum  plane  are  determined  by  means  of  the  level  and 
leveling  rod ;  and  the  horizontal  length  of  the  section  is 
measured  with  an  engineer's  chain  or  tape,  and  divided  into 
equal  parts,  usually  100  feet  in  length,  called  stations,  marked 
by  stakes  numbered  0,  1,  2,  3,  and  so  on. 

Where  the  ground  is  very  irregular,  it  may  be  necessary, 
besides  taking  sights  at  the  regular  stakes,  to  take  occasional 
sights  at  points  between  them.  If,  for  instance,  at  a  point 


LEVELING 


81 


40  feet  in  advance  of  stake  3  (Fig.  55)  there  is  a  sudden 
rise  or  fall  in  the  surface,  the  height  of  this  point  would  be 
determined  and  recorded  as  at  stake  3.40. 

The  readings  of  the  rod  are  ordinarily  taken  to  the  nearest 
tenth  of  a  foot,  except  on  bench  marks  and  points  called 
turning  points,  where  they  are  taken  to  thousandths  of  a  foot. 

A  turning  point  is  a  point  on  which  the  last  sight  is  taken 
just  before  changing  the  position  of  the  level,  and  the  first 
sight  from  the  new  position  of  the  instrument.  A  turning 
point  may  be  coincident  with  one  of  the  stakes,  but  must 
always  be  a  hard  point,  so  that  the  foot  of-  the  rod  may  stand 
at  the  same  level  for  both  readings. 


0 

1 

1 

t 

J^ 

^1 

0 

J.U 

\ 

11 

1 
1 

1 

5 

1 

1 

1 
1 

1 

,  1  — 

FIG.  55 

Field  Work.  To  explain  the  method  of  obtaining  the  field 
notes  necessary  for  making  a  profile,  let  0,  1,  2,  3,  •  •  -,  11 
(Fig.  55)  represent  a  portion  of  a  section  to  be  leveled  and 
plotted.  Establish  a  bench  mark  at  or  near  the  beginning  of 
the  line,  measure  the  horizontal  length  of  the  section,  and  set 
stakes  100  feet  apart,  numbering  them  0,  1,  2,  3,  and  so  on. 
Place  the  level  at  some  point,  as  between  2  and  3,  and  take 
the  reading  of  the  rod  on  the  bench  —  4.832.  Let  PP'  repre- 
sent the  datum  plane,  say  15  feet  below  the  bench  mark;  then 

15  +  4.832  =  19.832 

is  the  height  of  the  line  of  sight  AB,  called  the  height  of  the 
instrument,  above  the  datum  plane. 


82 


SURVEYING 


Now  take  the  reading  at  0  =  5.2  =  0  A,  and  subtract  the 
same  from  19.832,  which  leaves  14.6  =  0  P,  the  height  of  the 
point  0  above  the  datum  plane.  Next  take  sights  at  1,  2,  3, 
3.40,  and  4,  equal,  respectively,  to  3.7,  3.0,  5.1,  4.8,  and  8.3,  and 
subtract  the  same  from  19.832  ;  the  remainders  16.1,  16.8, 


J. 

sT 
i 

I 

i 

4 

t 

5 

•^6 

^^1 

11 

FIG.  56 

14.7,  15.0,  and  11.5  are  respective  heights  of  the  points  1,  2, 
3,  3.40,  and  4. 

Then,  as  it  is  necessary  to  change  the  position  of  the 
instrument,  select  a  point  in  the  neighborhood  of  4  suitable 
as  a  turning  point  (t.p.  in  the  figure),  and  take  a  careful 
reading  on  it  —  8.480 ;  subtract  this  from  19.832,  and  the 
remainder,  11.352,  is  the  height  of  the  turning  point. 

Now  carry  the  instrument  forward  to  a  new  position,  as 
between  5  and  6,  shown  in  the  figure,  while  the  rodman 
remains  at  t.p,\  take  a  second  reading  on  t.p.  =  4.102,  and  add 
it  to  11.352,  the  height  of  t.p.  above  PP';-  the  sum  15.454  is 
the  height  of  the  instrument  CD  in  its  new  position. 

Take  sight  upon  5,  6,  and  7,  equal,  respectively,  to  4.9,  2.8, 
and  0.904  ;  subtract  these  sights  from  15.454,  and  the  results 
10.6,  12.7,  and  14.550  are  the  heights  of  the  points  5,  6,  and 
7,  respectively. 

The  point  7,  being  suitable,  is  made  a  turning  point,  and 
the  instrument  is  moved  forward  to  a  point  between  9  and  10. 
The  sight  at  7  =  6.870,  added  to  the  height  of  7  gives  21.420 
as  the  height  of  the  instrument  EF  in  its  new  position.  The 


LEVELING 


83 


readings  at  8,  9,  10,  and  11,  which  are,  respectively,  5.4,  3.6, 
5.8,  and  9.0,  subtracted  from  21.420  give  the  heights  of  these 
points,  namely,  16.0,  17.8,  15.6,  and  12.4. 

Proceed  in  like  manner  until  the  entire  section  is  leveled, 
establishing  bench  marks  at  intervals  along  the  line  to  serve 
as  reference  points  for  future  operations.  The  bench  marks 
should  be  described  with  sufficient  minuteness  to  enable  any 
one  not  connected  with  the  survey  to  locate  them  easily  and 
unmistakably.  A  record  of  the  work  is  given  in  the  following 
table : 


STATION 

+  s. 

H.I. 

-s. 

H.S. 

REMARKS 

B 

4.832 

15.0 

Bench  on  rock  20  ft. 

0 

19.832 

5.2 

14.6 

south  of  0 

1 

3  7 

16.1 

2 

3.0 

16.8 

3 

5.1 

14.7 

3  to  3.40  turnpike  road 

3.40 

4.8 

15.0 

4 

8.3 

11.5 

t.p. 

4.102 

8.480 

11.352 

5 

15.454 

4.9 

10.6 

6 

2.8 

12.7 

7 

6.870 

0.904 

14.550 

8 

21.420 

5.4 

16.0 

9 

3.6 

17.8 

10 

5.8 

15.6 

11 

9.0 

12.4 

B 

Bench  on  oak  stump 

12 

27  ft.  KE.  of  12, 

etc. 

etc. 

The  first  column  contains  the  numbers  or  names  of  all  the 
points  on  which  sights  are  taken.  The  second  column  con- 
tains the  sight  taken  on  the  first  bench  mark,  and  the  sight 
on  each  turning  point  taken  immediately  after  the  instrument 


84  SURVEYING 

has  been  moved  to  a  new  position.  These  are  called  plus 
sights  (+  5.)  because  they  are  added  to  the  heights  of  the 
points  on  which  they  are  taken  to  obtain  the  height  of  the 
instrument  given  in  the  third  column  (#./.).  The  fourth 
column  contains  all  the  readings  except  those  recorded  in  the 
second  column.  These  are  called  minus  sights  (—  S.)  because 
they  are  subtracted  from  the  numbers  in  the  third  column  to 
obtain  all  the  numbers  in  the  fifth  column  except  the  first, 
which  is  the  assumed  depth  of  the  datum  plane  below  the 
bench.  The  fifth  column  (H.S.,  height  of  surface)  contains 
the  required  heights  of  all  the  points  of  the  section  named 
in  the  first  column  together  with  the  heights  of  all  benches 
and  turning  points. 

Making  the  Profile.  Draw  a  line  PP1  (Fig.  56),  to  repre- 
sent the  datum  plane,  and  beginning  at  some  point  as  P,  lay 
off  the  distances  100,  200,  300,  340,  400  feet,  and  so  on,  to  the 
right,  using  some  convenient  scale,  say  200  feet  to  the  inch. 
At  these  points  of  division  erect  perpendiculars  equal  in  length 
to  the  height  of  the  points  0,  1,  2,  3.40,  4,  •  •  •,  given  in  the 
fifth  column  of  the  above  field  notes,  using  in  this  case  a  larger 
scale,  say  20  feet  to  the  inch.  Through  the  extremities  of 
these  perpendiculars  draw  the  irregular  line  0,  1,  2,  3,  •  •  •,  11, 
and  the  result,  with  some  explanatory  figures,  is  the  required 
plot  or  profile. 

The  making  of  a  profile  is  much  simplified  by  the  use  of 
profile  paper,  which  may  be  had  by  the  yard  or  roll. 

If  a  horizontal  plot  is  required,  the  bearings  of  the  differ- 
ent portions  of  the  section  must  be  taken.  Such  a  plot  should 
be  made,  if  it  will  assist  in  properly  understanding  the  field 
work,  or  if  it  is  desirable  for  future  reference  in  connection 
with  the  field  notes.  Sometimes  both  the  profile  and  the  plot 
are  drawn  side  by  side  on  the  same  sheet ;  in  this  case,  if 
the  line  leveled  over  is  not  straight,  the  profile  will  be  longer 
than  the  plot. 


LEVELING 


85 


SECTION   XX 
TOPOGRAPHIC   LEVELING 

The  principal  object  of  topographic  surveying  is  to  show 
the  contour  of  the  ground.  This  operation,  called  topographic 
leveling,  is  performed  by  representing  on  paper  the  curved 
lines  in  which  parallel  horizontal  planes  at  uniform  distances 
from  each  other  would  meet  the  surface.  It  is  evident  that 
all  points  in  the  intersection  of  a  horizontal  plane  with  the 
surface  of  the  ground  are  at  the  same  level.  Hence,  it  is 
necessary  only  to  find  points  at  the  same  level  and  join  these 
to  determine  a  line  of  intersection. 

The  method  commonly  employed  will  be  understood  by 
reference  to  Fig.  57.  The  ground  ABCD  is  divided  into 
equal  squares,  and  a  numbered 
stake  driven  at  each  intersection. 
By  means  of  a  level  and  leveling 
rod  the  heights  of  the  other  sta- 
tions above  m  and  Z>,  the  lowest 
stations,  are  determined.  A  plot 
of  the  ground  with  the  intersect- 
ing lines  is  then  drawn,  and  the 
height  of  each  station  written  as 
in  the  figure. 

Suppose  that  the  horizontal 
planes  are  2  feet  apart ;  if  the 
first  passes  through  m  and  D,  the 

second  will  pass  through  p,  which  is  2  feet  above  m  ;  and 
since  n  is  3  feet  above  m,  the  second  plane  will  cut  the  line 
mn  in  a  point  s  determined  by  the  proportion  mn  :  ms  =  3:2. 
In  like  manner,  the  points  t,  q,  and  r  are  determined. 

The  irregular  line  tsp  -  •  •  qr  represents  the  intersection  of 
the  second  horizontal  plane  with  the  surface  of  the  ground. 


86  SURVEYING 

In  like  manner,  the  intersections  of  the  planes,  respectively, 
4,  6,  and  8  feet  above  m  are  traced.  The  more  rapid  the 
change  in  level  the  nearer  these  lines  approach  each  other. 

SECTION   XXI 
DRAINAGE   SURVEYING 

Preliminaries.  The  locality  to  be  drained  should  first  be 
carefully  reconnoitered,  with  the  view  of  ascertaining  the 
general  feature  of  the  land  so  as  to  enable  the  surveyor 
properly  to  locate  the  drains ;  the  beginning,  route,  and 
terminus  of  which  should  all  be  definitely  planned.  By  the 
beginning  of  a  drain  is  meant  its  highest  point. 

Field  Work.  The  field  work  is  essentially  the  same  for 
under  drains  and  for  open  drains.  The  first  thing  is  to 
establish  the  line  of  a  drain.  This  includes  the  setting  of 
stakes  at  intervals  of  from  50  feet  to  100  feet,  and  also 
wherever  there  is  an  angle  in  the  line ;  the  bearings  and 
lengths  of  the  successive  straight-line  sections,  beginning 
with  the  instrument  set  over  the  beginning  of  the  drain ; 
and  the  designation  by  distances  of  the  points  of  meeting 
of  roads  and  land  lines.  Levels  of  the  lines  are  then  taken 
in  accordance  with  the  method  described  on  pp.  81-83.  If 
circumstances  will  permit,  it  is  sometimes  of  advantage  to 
have  the  leveling  process  go  hand  in  hand  with  the  estab- 
lishing of  the  line. 

Plot  and  Profile.  If  a  considerable  region  is  to  be  drained, 
a  plot  should  be  made  of  the  entire  tract,  and  on  this  plot 
should  be  drawn,  in  proper  position,  the  lines  of  the  drain 
and  its  branches.  In  a  suitable  place  on  the  sheet  should  be 
noted  the  courses  of  the  various  sections  of  the  drain  and 
the  number  of  linear  feet  belonging  to  each  owner  of  land 
within  the  tract  drained.  A  profile  should  also  be  made, 


LEVELING 


87 


as  shown  on  page  84.  From  this  profile  inspection  will 
determine  whether  a  single  grade  will  suffice,  or  whether  a 
succession  of  different  grades  will  be  better. 


EXERCISE    IX 

1.  Find  the  difference  of  level  of  two  places  from  the  fol- 
lowing field  notes:  backsights,  5.2,  6.8,  and  4.0;  foresights, 
8.1,  9.5,  and  7.9. 

2.  Stake  0  of  the  following  notes  stands  at  the  lowest  point 
of  a  pond  to  be  drained  into  a  creek ;  stake  10  stands  at  the 
edge  of  the  bank,  and  10.25  at  the  bottom  of  the  creek.    Make 
a  profile,  draw  the  grade  line  through  0  and  10.25,  and  fill  out 
the  columns  H.G.  and  C.,  the  former  to  show  the  height  of 
grade  line  above  the  datum,  and  the  latter,  the  depth  of  cut 
at  the  several  stakes  necessary  to  construct  the  drain. 


STATION 

+  S. 

H.I. 

-s. 

H.S. 

H.G. 

C. 

REMARKS 

B 

6.000 

25 

Bench  on  rock 

0 

10.2 

20.8 

0.0 

30  ft.  west  of 

1 

5.3 

5.3 

stake  1 

2 

4.6 

3 

4.0 

4 

6.8 

5 

4.572 

7.090 

6 

3.9 

7 

2.0 

8 

4.9 

9 

4.3 

10 

4.5 

10.25 

11.8 

Horizontal  scale,    2  ch.  =  1  in. 
Vertical  scale,       20  ft.  =  1  in. 


88 


SURVEYING 


3.  Find  the    difference    in   altitude  between   the   highest 
point  and  the  lowest  point  of  the  campus  or  of  a  field. 

4.  Obtain  the  data  necessary  for  a  profile  of  a  half  mile  of 
highway,  and  make  the  profile. 

5.  Write  the  proper  numbers  in  the  third  and  fifth  columns 
of  the  following  table  of  field  notes,  and  make  a  profile  of  the 
section. 


STATION 

+  S. 

H.I. 

-s. 

H.S. 

REMARKS 

B 

0 

6.944 

7.4 

20 

Bench  on  post  22  ft. 
north  of  0 

1 

5.6 

2 

3.9 

3 

4.6 

*.*. 

4 

3.855 

5.513 
4.9 

5 

3.5 

6 

1.2 

CHAPTER   VI 
RAILROAD   SURVEYING 

SECTION   XXII 
LAYING   OUT   THE   ROUTE 

Preliminary  Survey.  After  it  has  been  decided  which  of 
several  feasible  lines  is  the  best,  a  preliminary  survey  for 
final  location  should  be  made.  This  should  include,  among 
other  things,  data  referring  to  elevations,  depressions,  streams 
to  be  crossed,  highways,  buildings  obstructing,  character  of 
soil,  and  natural  resources  affording  materials  for  construc- 
tion ;  also  data  referring  to  proximity  to  towns,  titles  of 
land,  rights  of  way,  and  so  on. 

Establishing  the  Roadbed.  When  the  general  route  of  a 
railroad  has  been  determined,  a  middle  surface  line  is  run 
with  the  transit.  A  profile  of  this  line  is  determined,  as 
on  page  84.  The  leveling  stations  are  commonly  1  chain 
(100  feet)  apart.  Places  of  different  level  are  connected  by 
a  gradient  line,  which  intersects  the  perpendiculars  to  the 
datum  line  at  the  leveling  stations  in  points  determined  by 
simple  proportion.  Hence,  the  distance  of  each  leveling 
station  above  or  below  the  level  or  gradient  line  which 
represents  the  position  of  the  roadbed  is  known. 

SECTION   XXIII 
CROSS-SECTION   WORK 

Excavations.  If  the  roadbed  lies  below  the  surface,  an 
excavation  is  made.  Let  ACBD  (Fig.  58)  represent  a  cross 

89 


90 


SURVEYING 


section  of  an  excavation,/  a  point  in  the  middle  surface  line, 
/'  the  corresponding  point  in  the  roadbed,  and  CD  the  width 
of  the  excavation  at  the  bottom.  The  slopes  at  the  sides  are 
commonly  made  so  that  A  A'  =  f  A'C,  and  BB'  =  f  DB'.  When 
ff  and  CD  are  known,  the  points  A,  B,  C1,  and  D'  are  readily 
determined  by  a  level  and  tape  measure. 


A' 


f 
FIG.  58 


If  from  the  area  of  the  trapezoid  ABB'A'  the  areas  of 
the  triangles  A  A'C  and  BB'D  are  deducted,  the  remainder 
is  the  area  of  the  cross  section.  In  like  manner  the  cross 
section  at  the  next  station  may  be  determined.  These  two 
cross  sections  are  the  bases  of  a  solid  whose  volume  will 
be  the  amount  of  the  excavation.  Since  the  cross  sections 
are  not  similar,  the  computations,  to  be  accurate,  should 
be  made  by  means  of  the  Prismoidal  Formula  (Geometry, 
§  733). 

Embankments.  If  the  roadbed  lies  above  the  surface,  an 
embankment  is  made,  the  cross  section  of  which  is  like  that 
of  the  excavation,  but  inverted. 


Fig.  59  represents  the  cross  section  of  an  embankment  which 
is  lettered  so  as  to  show  its  relation  to  the  excavation  of  Fig.  58. 


RAILROAD   SURVEYING 


91 


SECTION   XXIV 
CURVES 

Principles.  When  it  is  necessary  to  change  the  direction 
of  a  railroad  it  is  done  gradually  by  a  curve,  usually  the 
arc  of  a  circle.  Let  AF  and 
AO  (Fig.  60)  represent  two 
lines  to  be  thus  connected. 
Take  any  convenient  length 
AB  =  AE  =  t.  The  inter- 
section of  the  perpendicu-  F 
lars  EC  and  EC  determines 
the  centre  C,  and  the  radius 
of  curvature  EC  =  r.  The 
length  of  the  radius  de- 
pends on  the  angle  A  and  the  tangent  AB. 
triangle  ABC, 


FIG. 


For,  in  the  right 


tan  BA  C  — 


AB 


or  tan     A=- 


Hence, 


The  degree  of  a  railroad  curve  is  the  angle  subtended  at  the 
centre  of  the  curve  by  a  chord  of  100  feet.  If  D  is  the  degree 
of  a  curve  and  r  its  radius, 


.     .  50 

sin  4-  D  =  — •  3 
r 


and  r  =  50  esc  -J-  D. 


For  example,  a  6°  curve  has  a  radius  of  955.37  feet. 

Sometimes  the  topography  of  the  route  is  such  as  to  neces- 
sitate a  successive  series  of  curves  of  different  radii,  in  which 
case  the  whole  series  of  curves  is  called  a  compound  curve,  the 
principles  involved  being  the  same  for  each  component  as  for 
a  simple  curve. 


92 


SURVEYING 


Methods  of  laying  out  the  Curve.  1.  Let  Em  (Fig.  61) 
represent  a  portion  of  the  tangent.  It 
is  required  to  find  mP,  the  perpendicu- 
lar to  the  tangent  meeting  the  curve 
at  P. 

mP  =  En  =  CB  —  Cn. 


FIG.  61 


Cn  =  "V  CP2  -  Pn 


Hence, 

2.  It  is  required  to  find  mP  (Fig.  62) 
in  the  direction  of  the  centre. 

raP  =  mC  —  PC. 
But   mC  = 
Hence, 


FIG.  62 


3.  Place  transits  at  E  and  E  (Fig.  63).     Direct  the  tele- 
scope  of  the   former  to    E, 

and  of  the  latter  to  A.    Turn  A 

each  toward  the  curve  the 

same    number    of    degrees, 

and   mark  P,  the  point  of 

intersection  of  the  lines  of 

sight.    P  is  a  point  in  the 

circle  to  which  AB  and  AE 

are   tangents    at   B  and  E, 

respectively. 

4.  If  the  degree  D  of  the  curve  is  given  and  the  tangent 
BA  at  B  (Fig.  64),  place  the  transit  at  B  and  direct  the  tele- 
scope toward  A.    Turn  off  successively  the  angles  ABP,  PBP', 


FIG.  63 


RAILROAD    SURVEYING 


93 


P'BP",  •  •  •,  each  equal  to 
%D,  and  take  BP,  PP',  P'P", 

•  • .,  each  100  feet,  the  length 
of  the  tape.  Then,  P,  P', 
P",  •  •  •  lie  on  the  required 
curve. 

If  the  angle  A  and  the 
tangent  distance  BA  =  t 
are  given,  D  can  be  found 
from  the  formulas 

50 


FIG.  61 


sin  -J-  D  =  —  j  and  r  =  t  tan  %  A. 
50 


Whence,        sin  £  D  —  — 


EXERCISE    X 

1.  The  cross-section  areas  at  five  stations,  100  feet  apart, 
of  a  railroad  cut  are,  respectively,  576.8  square  feet,  695.1 
square  feet,  809.5  square  feet,  652.0  square  feet,  and  511.7 
square  feet.     Compute  the  volume  of  material  in  this  portion 
of  the  cut :   (i)  on  the  hypothesis  that  the  cross  sections  are 
similar ;   (ii)  on  the  hypothesis  that  they  are  dissimilar,  the 
alternate  cross  sections  being  regarded  as  mid-sections. 

2.  Find  the  radius  of  a  curve  of  1°,  of  2°,  of  3°,  of  4°,  of  5°. 

3.  Two  adjacent  straight  sections  of  a  railroad  form  an 
angle  of  148°  16'.     They  are  joined  by  a  curve  touching  each 
of  them  at  the  distance  of  388  feet  from  the  vertical  point. 
Find  the  radius  and  the  degree  of  the  curve. 

4.  Lay  out  a  curve  by  the  first  or  second  method,   and 
check  the  work  by  means  of  one  of  the  transit  methods. 


CHAPTER    VII 

CITY   SURVEYING 

SECTION  XXV 

FIELD    WORK 

Instruments.  Since  the  principles  in  city  surveying  are 
essentially  the  same  as  those  in  land  surveying,  instruments 
of  the  same  general  character  as  the  instruments  already 
described  may  be  used,  except  that  in  this  class  of  work  the 
ordinary  compass  and  the  chain  are  set  aside.  For  tho  smaller 
cities,  an  instrument  such  as  the  surveyor's  transit  is  suffi- 
cient in  accuracy  for  the  purposes  of  angle  measurement  and 
for  leveling.  However,  when  extreme  accuracy  is  demanded, 
as  in  the  case  of  large  cities,  specially  made  instruments 
should  be  used :  a  transit  reading  to  30  seconds,  or  even  to 
10  seconds ;  a  high-grade  Y  level  of  at  least  20-inch  length ; 
•and  a  standard  tape,  tested  for  sag  and  temperature. 

Streets.  In  most  cases  the  city  engineer  must  take  the 
streets  as  he  finds  them.  When  a  city  has  outgrown  its 
original  plan,  if  indeed  it  had  any,  sheer  necessity  may 
demand  the  location  of  additional  streets  or  changes  in 
existing  streets.  If  a  proposed  town  or  city  is  to  be  laid 
out,  the  general  contour  of  the  ground  and  location  of  the 
site  determine  to  a  great  extent  the  system  of  streets  to  be 
adopted.  Experience  has  shown  that  wherever  possible  a 
rectangular  system  of  street  lines,  with  a  few  well-located 
diagonal  streets,  is  the  most  satisfactory.  Streets  ordinarily 
vary  in  width  from  50  to  100  feet,  and  each  sidewalk  from  7 

04 


CITY   SURVEYING 


n 


to  15  feet.  The  principal  improvements  of  streets  are  grad- 
ing, paving,  setting  curbs,  laying  sidewalks,  constructing 
sewers,  and  laying  water  pipes. 


96  SURVEYING 

The  field  work  necessary  for  all  these  may  be  included 
under  the  heads  of  leveling,  locating  lines,  and  locating 
points,  which  have  already  been  described. 

Blocks  and  Lots.  There  is  no  established  rule  for  the 
size  of  either  blocks  or  lots.  Fig.  65  gives  some  idea  of  their 
dimensions.  The  location  of  a  block  is  described  by  refer- 
ence to  the  streets  which  bound  it.  A  lot  is  described  by 
number  and  block,  or  by  number  alone,  or  by  giving  the 
location  and  length  of  its  bounding  lines.  The  co-ordinate 
system  of  location  of  points,  described  on  page  53,  has 
much  in  its  favor  for  use  in  city  surveying.  Monuments 
at  points  of  reference  and  at  intersections  of  streets  and 
corners  of  lots  should  be  of  permanent  character,  and  set 
with  extreme  care. 

SECTION   XXVI 
OFFICE   WORK 

Plots.  Among  the  more  important  plots  that  should  be 
prepared  by  the  city  engineer  are  a  complete  city  map,  drawn 
to  scale,  showing  the  streets  and  alleys,  blocks  and  lots,  with 
dimensions,  and  the  location  of  railroads,  street-car  lines,  sew- 
age system,  water-pipe  lines,  and  so  on ;  a  topographical  map 
of  the  entire  city,  including  as  may  be  found  desirable  por- 
tions of  the  surrounding  region  ;  a  profile  map  of  the  streets. 
These  are  made  from  the  field  notes,  which  should  be  amply 
and  carefully  prepared. 

Records.  No  work  of  importance,  whether  done  in  the  field 
or  in  the  office,  should  fail  to  be  recorded  in  some  perma- 
nent form.  Field  notes,  computations,  plots,  and  copies  of 
work  specially  prepared  should  be  properly  indexed  and  filed 
away. 


ANSWERS  97 


SURVEYING 


Exercise  II.  Page  22 

2.    540°.  4.    N.  51°  30'  E. 

Exercise  III.  Page  27 

2.    360°.  3.    235  ft.  3.8  in. 

Exercise  V.  Page  55 


1.  8  A.  64  P. 

6.  13  A.  6TV  P. 

11.  4  A.  35  P. 

2.  16  A.  74|f  P. 

7.  2  A.  581  p. 

12.  4  A.  110  P. 

3.  4  A.  5sV  P. 

8.  11  A.  157  P. 

13.  6  A.  23i|  P. 

4.  115oV  p- 

9.  7.51925. 

5.  3  A.  78  P. 

10.  13.0735. 

Exercise  VI.     Page  64 

1.  2  A.  26  P.  3.    8  A.  54  P.  5.    2  A.  78  P.  7.    5  A.  42  p. 

2.  20  A.  12  P.          4.    3  A.  122  p.          6.    6  A.  2  p.  8.    2  A.  151  p. 

Exercise  VII.    Page  66 
1.    2  A.  121  p.  2.    98  A.  92  p. 

Exercise  VIII.    Page  69 

1.  J.#=3.75ch.  4.    AE=5.5Qch. 

2.  ^#  =  3.50ch.;  5.    CE  =  4.456  ch. 

EG  =  3.42  ch.  6     AD  _  2.275  ch. ;   BE  =  1.82  ch. 

3.  ^  =  4.55ch.  7.    AD  =  4. 51  ch.;  £#  =  3.61  ch. 

8.  The  distances  on  AB  are  2  ch.,  3  ch.,  and  5  ch. 

9.  EM  (on  EA)  =  2.5087  ch. ;  AN  (on  AB)  =  6.4390  ch. 


98 


SURVEYING 


10.  LetEG>DF;  then  AE  =  12.247  ch.,  AG  =  9.798  ch.,  AD  =  8.660 

ch.,  AF=  6.928  ch. 

11.  LetDG>EF-  then  CG=  14.862  ch.,  CD  =  13.113  ch.,  CF=  11.404 

ch.,  CE  =  10.062  ch. 

Exercise  IX.     Page  87 

1.  9.5. 

2.  Column  H.G.  20.8,  20.4,  20.0,  19.6,  19.2,  18.8,   18.4,   18.0,   17.6, 

17.2,  16.8,  16.7. 
Column  C.  0.0,  5.3,  6.4,  7.4,  5.0,  5.1,  6.2,  8.5,  6.0,  7.0,  7.2,  0.0. 


9       10  10.25 


5.    Third  column  :   26.944  opposite  0;  25.286  opposite  4. 

Fifth  column  :   20,  19.5,  21.3,  23,  22.3,  21.431,  20.4,  21.8,  24.1. 


Exercise  X.    Page  93 

1.  9986. 5  cu.  yd.;  9994.9  cu.  yd. 

2.  5730ft.;  2865ft.;  1910ft.;  1433ft.;  1146ft. 

3.  1365ft.;  4°  11' 53". 


FIVE  -PLACE 


LOGARITHMIC  AND  TRIGONOMETRIC 


TABLES 


ARRANGED    BY 


G.    A.   WENTWORTH,   A.M. 


AND 


G.   A.   HILL,   A.M. 


GINN  &  COMPANY 

BOSTON  •  NEW  YORK  .  CHICAGO  .  LONDON 


Entered  according  to  Act  of  Congress,  in  the  year  1882,  by 

G.  A.  WENTWORTH  AND  G.  A.  HILL 
in  the  office  of  the  Librarian  of  Congress  at  Washington 


Copyright.  1895,  by  G.  A.  WENTWORTH  and  G.  A.  HILL. 

210.2 


Scltfjenaum 


GIXN    &   COMPANY.  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


INTRODUCTION. 


1.  If  the  natural  numbers  are  regarded  as  powers  of  ten,  the 
exponents  of  the  powers  are  the  Common  or  Briggs  Logarithms  of 
the  numbers.     If  A  and  B  denote  natural  numbers,  a  and  b  their 
logarithms,  then  10a  —  A,  10&  —  B ;  or,  written  in  logarithmic  form, 

log  A  =  a,         log  B  =  b. 

2.  The  logarithm  of  a  product  is  found  by  adding  the  logarithms 
of  its  factors. 

For,  AX  B  =  10«  X  10&  =  10«  +  6. 

Therefore,          log  ( A  X  B)  =  a  +  b  =  log  A  +  log  B. 

3.  The  logarithm  of  a  quotient  is   found  by   subtracting  the 
logarithm  of  the  divisor  from  that  of  the  dividend. 

l=i!=-->- 

Therefore,  log  —  =  a  —  b  =  log  A  —  log  B. 

_D 

4.  The  logarithm  of  a  power  of  a  number  is  found  by  multiply- 
ing the  logarithm  of  the  number  by  the  exponent  of  the  power. 

For,  An  =  (10°)»  =  10an. 

Therefore,          log-4n  =  an  =  n  log  A. 

5.  The  logarithm  of  the  root  of  a  number  is  found  by  dividing 
the  logarithm  of  the  number  by  the  index  of  the  root. 

n, —         n, ? 

For,  VZ  =  VlO"  —  10". 

Therefore,          log  tfZ  =  S  =  !°£4. 
n          n 

6.  The  logarithms  of  1,  10,  100,  etc.,  and  of  0.1,  0.01,  0.001, 
etc.,  are  integral  numbers.     The  logarithms  of  all  other  numbers 
are  fractions. 


IV  LOGARITHMS. 

For,    10°  =        1,  hence        log  1  =  0 ;  10-1  =      0.1,  hence      log  0.1  =  —  1  • 

101  =      10,  hence      log  10  =  1 ;  10~2  =    0.01,  hence    log  0.01  =  —  2  ; 

102  =    100,  hence    log  100  =  2  ;  10-3  =  0.001,  hence  log  0.001  =  -  3  ; 

103  =  1000,  hence  log  1000  =  3 ;  and  so  on. 

If  the  number  is  between       1  and  10,  the  logarithm  is  between      0  and      1. 

If  the  number  is  between     10  and  100,  the  logarithm  is  between      1  and      2. 

If  the  number  is  between   100  and  1000,  the  logarithm  is  between      2  and      3. 

If  the  number  is  between       land  0.1,  the  logarithm  is  between      0  and  —  1. 

If  the  number  is  between  0.1  and  0.01,  the  logarithm  is  between  — 1  and  — 2. 
If  the  number  is  between  0.01  and  0.001,  the  logarithm  is  between  —2  and  —3. 
And  so  on. 

7.  If  the  number  is  less  than  1,  the  logarithm  is  negative  (§  6), 
but  is  written  in  such  a  form  that  the  fractional  part  is  always  positive. 

For  the  number  may  be  regarded  as  the  product  of  two  factors,  one  of 
which  lies  between  1  and  10,  and  the  other  is  a  negative  power  of  10 ;   the 
logarithm  will  then  take  the  form  of  a  difference  whose  minuend  is  a  positive 
proper  fraction,  and  whose  subtrahend  is  a  positive  integral  number. 
Thus,  0.48  =  4.8X0.1. 

Therefore  (§  2),  log     0.48  =  log  4.8  +  log  0.1  =  0.68124  -  1.     (Page  1.) 
Again,  0.0007  =  7  X  0.0001. 

Therefore,  log  0.0007  =  log  7  +  log  0.0001  =  0.84510  —  4. 

The  logarithm  0.84510  —  4  is  often  written  4.84510. 

8.  Every  logarithm,  therefore,  consists  of  two  parts  :  a  positive 
or  negative  integral  number,  which  is  called  the  Characteristic,  and 
a  positive  proper  fraction,  which  is  called  the  Mantissa. 

Thus,  in  the  logarithm  3.52179,  the  integral  number  3  is  the  characteristic, 
and  the  fraction  .52179  the  mantissa.  In  the  logarithm  0.78254  —  2,  the  inte- 
gral number  —  2  is  the  characteristic,  and  the  fraction  0.78254  is  the  mantissa. 

9.  If  the  logarithm  is  negative,  it  is  customary  to  change  the 
form  of  the  difference  so  that  the  subtrahend  shall  be  10  or  a  multiple 
of  10.     This  is  done  by  adding  to  both  minuend  and  subtrahend  a 
number  which  will  increase  the  subtrahend  to  10  or  a  multiple  of  10. 

Thus,  the  logarithm  0.78254  —  2  is  changed  to  8.78254  —  10  by  adding  8  to 
both  minuend  and  subtrahend.  The  logarithm  0.92737  —  13  is  changed  to 
7.92737  —  20  by  adding  7  to  both  minuend  and  subtrahend. 

10.  The  following  rules  are  derived  from  §  6 :  — 

If  the  number  is  greater  than  1,  make  the  characteristic  of  the 
logarithm  one  unit  less  than  the  number  of  figures  on  the  left  of 
the  decimal  point. 

If  the  number  is  less  than  1,  make  the  characteristic  of  the  loga- 
rithm negative,  and  one  unit  more  than  the  number  of  zeros  between 
the  decimal  point  and  the  first  significant  figure  of  the  given  number. 


INTRODUCTION.  V 

If  the  characteristic  of  a  given  logarithm  is  positive,  make  the 
number  of  figures  in  the  integral  part  of  the  corresponding  number 
one  more  than  the  number  of  units  in  the  characteristic. 

If  the  characteristic  is  negative,  make  the  number  of  zeros  between 
the  decimal  point  and  the  first  significant  figure  of  the  correspond- 
ing number  one  less  than  the  number  of  units  in  the  characteristic. 

Thus,  the  characteristic  of  log  7849.27  =  3  ; 

the  characteristic  of  log  0.037  =  —  2  =  8.00000  —  10. 

If  the  characteristic  is  4,  the  corresponding  number  has  five  figures  in  its  inte- 
gral part.  If  the  characteristic  is  —  3,  that  is,  7.00000  —  10,  the  corresponding 
fraction  has  two  zeros  between  the  decimal  point  and  the  first  significant  figure. 

11.  The  logarithms  of  numbers  that  can  be  derived  one  from 
another  by  multiplication  or  division  by  an  integral  power  of  10 
have  the  same  mantissa. 

For,  multiplying  or  dividing  a  number  by  an  integral  power  of  10  will 
increase  or  diminish  its  logarithm  by  the  exponent  of  that  power  of  10 ;  and 
since  this  exponent  is  an  integer,  the  mantissa  of  the  logarithm  will  be 
unaffected. 

Thus,        log  4.6021      =0.66296.     (Page  9.) 

log  460.21      =  log  (4.6021  X  102)  =  log  4.6021  +  log  102 

=  0.66296  +  2  =  2.66296. 
log  460210     =  log  (4.6021  X  105)  =  log  4.6021  +  log  106 

=  0.66296  +  5  =  5.66296. 
log  0.046021  =  log  (4.6021  -=-  10*)  =  log  4.6021  -  log  102 

=  0.66296  —  2  =  8.66296  -  10. 


TABLE   I. 

12.  In  this  table  (pp.  1-19)  the  vertical  columns  headed  N  con- 
tain the  numbers,  and  the  other  columns  the  logarithms.  On  page  1 
both  the  characteristic  and  the  mantissa  are  printed.  On  pages 
2-19  the  mantissa  only  is  printed. 

The  fractional  part  of  a  logarithm  can  be  expressed  only  approx- 
imately, and  in  a  five-place  table  all  figures  that  follow  the  fifth  are 
rejected.  Whenever  the  sixth  figure  is  5,  or  more,  the  fifth  figure 
is  increased  by  1.  The  figure  5  is  written  when  the  value  of  the 
figure  in  the  place  in  which  it  stands,  together  with  the  succeeding 
figures,  is  more  than  4-J-,  but  less  than  5. 

Thus,  if  the  mantissa  of  a  logarithm  written  to  seven  places  is  5328732,  it  is 
written  in  this  table  (a  five-place  table)  53287.  If  it  is  5328751,  it  is  written 
53288.  If  it  is  5328461  or  5328499,  it  is  written  in  this  table  53285.. 

Again,  if  the  mantissa  is  5324981,  it  is  written  53250 ;  and  if  it  is  4999967,  it 
is  written  50000. 


Vi  LOGARITHMS. 

This  distinction  between  5  and  5,  in  case  it  is  desired  to  curtail 
still  further  the  mantissas  of  logarithms,  removes  all  doubt  whether 
a  5  in  the  last  given  place,  or  in  the  last  but  one  followed  by  a 
zero,  should  be  simply  rejected,  or  whether  the  rejection  should 
lead  us  to  increase  the  preceding  figure  by  one  unit. 

Thus,  the  mantissa  13925.  when  reduced  to  four  places  should  be  1392  ;  but 
13925  should  be  1393. 

To  FIND  THE  LOGARITHM  OF  A  GIVEN  NUMBER. 

13.  If  the  given  number   consists   of   one    or   two  significant 
figures,  the  logarithm  is   given  on  page  1.      If  zeros  follow  the 
significant  figures,  or  if  the  number  is  a  proper  decimal  fraction, 
the  characteristic  must  be  determined  by  §  10. 

14.  If  the  given  number  has  three  significant  figures,  it  will  be 
found  in  the  column  headed  1ST  (pp.  2-19),  and  the  mantissa  of  its 
logarithm  in  the  next  column  to  the  right,  and  on  the  same  line. 
Thus, 

Page    2.     log  145  =  2. 16137,  log  14500  =  4.16137. 

Page  14.     log  716  =  2.85491,  log  0.716  =  9.85491  —  10. 

15.  If  the  given  number  has  four  significant  figures,  the  first 
three  will  be  found  in  the  column  headed  N,  and  the  fourth  at  the 
top  of  the  page  in  the  line  containing  the  figures  1,  2,  3,  etc.     The 
mantissa  will  be  found  in  the  column  headed  by  the  fourth  figure, 
and  on  the  same  line  with  the  first  three  figures.     Thus, 

Page  15.     log  7682    =  3.88547,         log  76.85    =  1.88564. 
Page  18.     log  93280  =  4.96979,         log  0.9468  =  9.97626  —  10. 

16.  If  the  given  number  has  five  or  more  significant  figures,  a 
process  called  interpolation  is  required. 

Interpolation  is  based  on  the  assumption  that  between  two  con- 
secutive mantissas  of  the  table  the  change  in  the  mantissa  is  directly 
proportional  to  the  change  in  the  number. 

Required  the  logarithm  of  34237. 

The  required  mantissa  is  (§  11)  the  same  as  the  mantissa  for  3423.7  ;  mere- 
fore  it  will  be  found  by  adding  to  the  mantissa  of  3423  seven-tenths  of  the 
difference  between  the  mantissas  for  3423  and  3424. 

The  mantissa  for  3423  is  53441. 

The  difference  between  the  mantissas  for  3423  and  3424  is  12. 

Hence,  the  mantissa  for  3423.7  is  53441  +  (0.7  X  12)  =  5344° 

Therefore,  the  required  logarithm  of  34237  is  4.53449. 


INTRODUCTION.  Vll 

Required  the  logarithm  of  0.0015764. 

The  required  mantissa  is  the  same  as  the  mantissa  for  1576.4 ;  therefore 
it  will  be  found  by  adding  to  the  mantissa  for  1576  four-tenths  of  the  difference 
between  the  mantissas  for  1576  and  1577. 

The  mantissa  for  1576  is  19756. 

The  difference  between  the  mantissas  for  1576  and  1577  is  27. 

Hence,  the  mantissa  for  1576.4  is  19756  +  (0.4  X  27)  —  19767. 

Therefore,  the  required  logarithm  of  0.0015764  is  7.19767  —  10. 

Required  the  logarithm  of  32.6708. 

The  required  mantissa  is  the  same  as  the  mantissa  for  3267.08;  therefore 
it  will  be  found  by  adding  to  the  mantissa  for  3267  eight-hundredths  of  the 
difference  between  the  mantissas  for  3267  and  3268. 

The  mantissa  for  3267  is  51415. 

The  difference  between  the  mantissas  for  3267  and  3268  is  13. 

Hence,  the  mantissa  for  3267.08  is  51415  +  (0.08  X  13)  =  51416. 

Therefore,  the  required  logarithm  of  32.6708  is  1.51416. 

17.  When  the  fraction  of  a  unit  in  the  part  to  be  added  to  the 
mantissa  for  four  figures  is  less  than  0.5  it  is  to  be  neglected ;  when 
it  is  0.5  or  more  than  0.5  it  is  to  be  taken  as  one  unit. 

Thus,  in  the  first  example,  the  part  to  be  added  to  the  mantissa  for  3423  is 
8.4,  and  the  .4  is  rejected.  In  the  second  example,  the  part  to  be  added  to  the 
mantissa  for  1576  is  10.8,  and  11  is  added. 


To  FIND    THE  ANTILOGARITHM  ;    THAT  is,   THE  NUMBER  CORRE- 
SPONDING TO  A  GIVEN  LOGARITHM. 

18.  If  the  given  mantissa  can  be  found  in  the  table,  the  first 
three  figures  of  the  required  number  will  be  found  in  the  same  line 
with  the  mantissa  in  the  column  headed  N,  and  the  fourth  figure  at 
the  top  of  the  column  containing  the  mantissa. 

The  position  of  the  decimal  point  is  determined  by  the  charac- 
teristic (§  10). 

Find  the  number  corresponding  to  the  logarithm  0.92002. 

Page  16.     The  number  for  the  mantissa  92002  is  8318. 

The  characteristic  is  0 ;   therefore,  the  required  number  is  8.318. 

Find  the  number  corresponding  to  the  logarithm  6.09167. 

Page  2.     The  number  for  the  mantissa  09167  is  1235. 

The  characteristic  is  6 ;  therefore,  the  required  number  is  1235000 

Find  the  number  corresponding  to  the  logarithm  7.50325  — 10. 

Page  6.     The  number  for  the  mantissa  50325  is  3186. 

The  characteristic  is  —  3 ;   therefore,  the  required  number  is  0.003186. 


Vlll  LOGARITHMS. 

19.  If  the  given  mantissa  cannot  be  found  in  the  table,  find 
in  the  table  the  two  adjacent  mantissas  between  which  the  given 
mantissa  lies,  and  the  four  figures  corresponding  to  the  smaller  of 
these  two  mantissas  will  be  the  first  four  significant  figures  of  the 
required  number.     If  more  than  four  figures  are  desired,  they  may 
be  found  by  interpolation,  as  in  the  following  examples : 

Find  the  number  corresponding  to  the  logarithm  1.48762. 

Here  the  two  adjacent  mantissas  of  the  table,  between  which  the  given  man- 
tissa 48762  lies,  are  found  to  be  (page  6)  48756  and  48770.  The  corresponding 
numbers  are  3073  and  3074.  The  smaller  of  these,  3073,  contains  the  first  four 
significant  figures  of  the  required  number. 

The  difference  between  the  two  adjacent  mantissas  is  14,  and  the  difference 
between  the  corresponding  numbers  is  1. 

The  difference  between  the  smaller  of  the  two  adjacent  mantissas,  48756, 
and  the  given  mantissa,  48762,  is  6.  Therefore,  the  number  to  be  annexed  to 
3073  is  T6¥  of  1  =  0.428,  and  the  fifth  significant  figure  of  the  required  number 
is  4. 

Hence,  the  required  number  is  30.734. 

Find  the  number  corresponding  to  the  logarithm  7.82326  — 10. 

The  two  adjacent  mantissas  between  which  82326  lies  are  (page  13)  82321 
and  82328.  The  number  corresponding  to  the  mantissa  82321  is  6656. 

The  difference  between  the  two  adjacent  mantissas  is  7,  and  the  difference 
between  the  corresponding  numbers  is  1. 

The  difference  between  the  smaller  mantissa,  82321,  and  the  given  mantissa, 
82326,  is  5.  Therefore,  the  number  to  be  annexed  to  6656  is  f  of  1  =  0.7,  and 
the  fifth  significant  figure  of  the  required  number  is  7. 

Hence,  the  required  number  is  0.0066567. 

In  using  a  five-place  table  the  numbers  corresponding  to  man- 
tissas may  be  carried  to  five  significant  figures,  and  in  the  first 
part  of  the  table  to  six  figures.* 

20.  The  logarithm  of  the  reciprocal  of  a  number  is  called  the 
Cologarithm  of  the  number. 

If  A  denotes  any  number,  then 

colog  A  =  log  —  =  log  1  —  log  A  (§  3)  =  —  log  A. 
A 

Hence,  the  cologarithm  of  a  number  is  equal  to  the  logarithm  of 
the  number  with  the  minus  sign  prefixed,  which  sign  affects  the 
entire  logarithm,  both  characteristic  and  mantissa. 

*In  most  tables  of  logarithms  proportional  parts  are  given  as  an  aid  to 
interpolation  ;  but,  after  a  little  practice,  the  operation  can  be  performed  nearly 
as  rapidly  without  them.  Their  omission  allows  a  page  with  larger-faced  type 
and  more  open  spacing,  and  consequently  less  trying  to  the  eyes. 


INTRODUCTION.  IX 

In  order  to  avoid  a  negative  mantissa  in  the  cologarithm,  it  is 
customary  to  substitute  for  —  log  A  its  equivalent 

(10  -  log  A)  — 10. 

Hence,  the  cologarithm  of  a  number  is  found  by  subtracting  the 
logarithm  of  the  number  from  10,  and  then  annexing  — 10  to  the 
remainder. 

The  best  way  to  perform  the  subtraction  is  to  begin  on  the  left 
and  subtract  each  figure  of  log  A  from  9  until  we  reach  the  last 
significant  figure,  which  must  be  subtracted  from  10. 

If  log  A  is  greater  in  absolute  value  than  10  and  less  than  20, 
then  in  order  to  avoid  a  negative  mantissa,  it  is  necessary  to  write 
—  log  A  in  the  form 

(20  —  log  A)  —  20. 

So  that,  in  this  case,  colog  A  is  found  by  subtracting  log  A  from 
20,  and  then  annexing  —  20  to  the  remainder. 

Find  the  cologarithm  of  4007. 

10  —10 

Page  8.  log  4007  =    3.60282 

colog  4007=    6.39718-10 

Find  the  cologarithm  of  103992000000. 

20  -20 

Page  2.     log  103992000000  =  11.01700 

colog  103992000000  =    8.98300  —  20 

If  the  characteristic  of  log  A  is  negative,  then  the  subtrahend, 
— 10  or  —  20,  will  vanish  in  finding  the  value  of  colog  A. 

Find  the  cologarithm  of  0.004007. 

10  -10 

log  0.004007  =    7.60282  -  10 
colog  0.004007  =    2.39718 

With  practice,  the  cologarithm  of  a  number  can  be  taken  from 
the  table  as  rapidly  as  the  logarithm  itself. 

By  using  cologarithms  the  inconvenience  of  subtracting  the  log- 
arithm of  a  divisor  is  avoided.  For  dividing  by  a  number  is 
equivalent  to  multiplying  by  its  reciprocal.  Hence,  instead  of 
subtracting  the  logarithm  of  a  divisor  its  cologarithm  may  be 
added. 


LOGARITHMS. 


EXERCISES. 


Find  the  logarithms  of : 


1.  6170. 

2.  0.617. 

3.  2867. 


4.  85.76. 

5.  296.8. 

6.  7004. 


7.  0.8694. 

8.  0.5908. 

9.  73243. 


10.  67.3208. 

11.  18.5283. 

12.  0.0042003. 


Find  the  cologarithms  of  : 


13.  72433. 

14.  802.376. 

15.  15.7643. 


16.  869.278. 

17.  154000. 

18.  70.0426. 


19.  0.002403. 

20.  0.000777. 

21.  0.051828. 


Find  the  antilogarithms  of  : 


22.  2.47246. 

23.  7.89081. 

24.  2.91221. 


25.  1.26784. 

26.  3.79029. 

27.  5.18752. 


28.  9.79029—10. 

29.  7.62328-10. 

30.  6.15465  —  10. 


COMPUTATION  BY  LOGARITHMS. 

21.    (1)  Find  the  value  of  x,  if  x  =  72214  X  0.08203. 

Page  14.  log  72214     =  4.85862 

Page  16.  log  0.08203  =  8.91397  -  10 

By  §  2.  logx  =3.77259 

Page  11.  x  =  5923.63 


(2)  Find  the  value  of  x,  if  x  =  5250  -j-  23487. 

Page  10.  log  5250  =  3.72016 

Page  4.  colog  23487  =  5.62917  -  10 

Page  4.  logx  =  9.34933  -  10  =  log  0.2235S 

.-.  x  =  0.22353 


/ox  v   j  4.1,        ^        f      -f  7.56X4667X567 

(3)  Find  the  value  of  x,  if  «  =  399^  X0.oo337  X  23435* 

Page  15.  log  7.56  =  0.87852 

Page  9.  log  4667  =  3.66904 

Page  11.  log  567  =  2.75358 

Page  17.  colog  899.1  =  7.04619  —  10 

Page  6.  .    colog  0.00337  >•=  2.47237 

Page  4.  colog  23435  =  5.63013  —  10 

Page  6.  log  x  —  2.44983  =  log  281.73 

,:x  =281.73. 


INTRODUCTION.  Xl 

C4)  Find  the  cube  of  376. 

Page  7.  log  376  =  2.57519 

Multiply  by  3  (§  4),  3 

Page  10.  log  3763  =  7.72557  =  log  53158600 

.-.  3763  =  53158600. 

(5)  Find  the  square  of  0.003278. 

Page  6.  log  0.003278  =    7.51561-10 

2 

Page  2.  log  0.0032782  =  15.03122  -  20  =  log  0.000010745 

.-.  0.0032782  =    0.000010745. 

(6)  Find  the  square  root  of  8322. 

Page  16.  log  8322  =  3.92023 

Divide  by  2  (§  5),  2)3.92023 

log  V8322  =  1.96012  =  log  91.226 

.-.  V8322  =  91.226. 

If  the  given  number  is  a  proper  fraction,  its  logarithm  will  have 
as  a  subtrahend  10  or  a  multiple  of  10.  In  this  case,  before  divid- 
ing the  logarithm  by  the  index  of  the  root,  both  the  subtrahend  and 
the  number  preceding  the  mantissa  should  be  increased  by  such  a 
number  as  will  make  the  subtrahend,  when  divided  by  the  index  of 
the  root,  10  or  a  multiple  of  10. 

(7)  Find  the  square  root  of  0.000043641. 

Page  8.  log  0.000043641      =    5.63989-10 

10  — 10 

Divide  by  2  (§  5),      2)15.63989-20 

Page  13.  log  VO. 000043641  =    7.81995  —  10  =  Log  0.0066062 

.-.  V0.000043641  =  0.0066062. 

(8)  Find  the  sixth  root  of  0.076553. 

'Page  15.  log  0.076553  =    8.88397-10 

50  -50 

Divide  by  6  (§  5),  6)58.88397-60 

Page  13.  log  VO. 076553        =    9.81400  -  10  =•  log  0-65163 

.-.  \/0. 076553       =    0.65163. 

EXERCISES. 

Find  by  logarithms  the  value  of : 

1     45607  5.6123  2.567 

'    31045'  A    0.01987'  0.05786* 


LOGARITHMS. 
0.06547 


'    74.938  X  0.05938 
4.657  X  0.03467 


3.908  X  0.07189 

0.0075389  X  0.0079 
0.00907  X  0009784* 

312  X  7.18  X  31.82 
519  X  8.27  X  5.132* 

0.007  X  57.83  X  28.13 
9.317  X  00.28  X  476.5* 

5.55  X  0.0007632  X  0.87654 
2.79  X  0.0009524  X  1.46785 


/0. 003457  X  43.387  X  99.  S 
\  0.005824  X  15.724  X  1.3* 


2  X  0.00025 
38  X  0.00089 


8/23.815  X  29.36  X  0.007  X  0.62487 
'  \ 0.00072  X  9.236  X  5.924  X  3.0007 


/3.1416  X  0.0314 
'  \  1.7285  X  0.01 75 


031416  X  0.0031416 


017285  X  0.0017285 


TABLE  II. 

22.    This  table  (page  20)  contains  the  value  of  the  number  TT, 
its  most  useful  combinations,  and  their  logarithms. 

Find  the  length  of  an  arc  of  47°  32'  57"  in  a  unit  circle. 

47°  32' 57"  =171177" 
log  171177  =  5.23344 

log  4>  =  4.68557  -  10 


log  arc  47°  32'  57"  =  9.91901  -  10  =  log  0.82994 
.-.  length  of  arc       =  0.82994. 

Find  the  angle  if  the  length  of  its  arc  in  a  unit  circle  =  0.54936. 
log  0. 54936  =  9. 73986  -  10 

colog  —  =  log  a"  =  5.31443 

log  angle  =  5.05429  =  log  113316 

.-.  angle  =  113316"  =  31°  28"  36". 


INTRODUCTION. 


23.  The  relations  between  arcs  and  angles  given  in  Table  II. 
are  readily  deduced  from  the  circular  measure  of  an  angle. 

In  Circular  Measure  an  angle  is  defined  by  the  equation 

arc 

angle  =  —  r=  —  > 
radius 

in  which  the  word  arc  denotes  the  length  of  the  arc  corresponding 
to  the  angle,  when  both  arc  and  radius  are  expressed  in  terms  of 
the  same  linear  unit. 

Since  the  arc  and  radius  for  a  given  angle  in  different  circles 
vary  in  the  same  ratio,  the  value  of  the  angle  given  by  this  equa- 
tion is  independent  of  the  value  of  the  radius. 

The  angle  which  is  measured  by  a  radius-arc  is  called  a  Radian, 
and  is  the  angular  unit  in  circular  measure. 

C  4-  C 

Since  C  =  2  irR,  it  follows  that  —  =  2  TT,  and  ^  =  TT.  Therefore, 

H  H 

If  the  arc  =  circumference,  the  angle  =  2  TT. 

If  the  arc  =  semicircumference,  the  angle  =  TT. 

If  the  arc  =  quadrant,  the  angle  =  %  TT. 

If  the  arc  =  radius,  the  angle  =  1. 

Therefore,  TT  =  180°,  £  TT  =  90°,  £  TT  =  60°,  £  TT  =  45°,  J  TT  =  30", 
£7T  =  22i°,  and  so  on. 

Since  180°  in  common  measure  equals  TT  units  in  circular  measure, 

1°  in  common  measure        =  T-^-  units  in  circular  measure  ; 

loO 

180°  . 
1  unit  in  circular  measure  =  -  •  in  common  measure. 

7T 

By  means  of  these  two  equations,  the  value  of  an  angle  expressed 
in  one  measure  may  be  changed  to  its  value  in  the  other  measure. 
Thus,  the  angle  whose  arc  is  equal  to  the  radius  is  an  angle  of 

180° 
1  unit  in  circular  measure,  and  is  equal  to  -  ,  or  57°  17'  45", 

very  nearly. 

TABLE   III. 

24.  This  table  (pp.  21-49)  contains  the  logarithms  of  the  trigo- 
nometric functions  of  angles.     In  order  to  avoid  negative  character- 
istics, the  characteristic  of  every  logarithm  is  printed  10  too  large. 
Therefore,  —  10  is  to  be  annexed  to  each  logarithm. 

On  pages  28-49  the  characteristic  remains  the  same  throughout 
each  column,  and  is  printed  at  the  top  and  the  bottom  of  the  column, 


XIV  LOGARITHMS. 

But  on  pp.  '60,  49,  the  characteristic  changes  one  unit  in  value  at  the 
places  marked  with  bars.  Above  these  bars  the  proper  characteristic 
is  printed  at  the  top,  and  below  them  at  the  bottom,  of  the  column. 

25.  On  pages  28-49  the  log  sin,  log  tan,  log  cot,  and  log  cos,  of 
1°  to  89°,  are  given  to  every  minute.  Conversely,  this  part  of  the 
table  gives  the  value  of  the  angle  to  the  nearest  minute  when 
log  sin,  log  tan,  log  cot,  or  log  cos  is  known,  provided  log  sin  or 
log  cos  lies  between  8.24186  and  9.99993,  and  log  tan  or  log  cot 
lies  between  8.24192  and  11.75808. 

If  the  exact  value  of  the  given  logarithm  of  a  function  is  not 
found  in  the  table,  the  value  nearest  to  it  is  to  be  taken,  unless 
interpolation  is  employed  as  explained  in  §  26. 

If  the  angle  is  less  than  45°  the  number  of  degrees  is  printed  at 
the  top  of  the  page,  and  the  number  of  minutes  in  the  column  to 
the  left  of  the  columns  containing  the  logarithm.  If  the  angle 
is  greater  than  45°,  the  number  of  degrees  is  printed  at  the  bottom 
of  the  page,  and  the  number  of  minutes  in  the  column  to  the  right 
of  the  columns  containing  the  logarithms. 

If  the  angle  is  less  than  45°,  the  names  of  its  functions  are 
printed  at  the  top  of  the  page ;  if  greater  than  45°,  at  the  bottom 
of  the  page.  Thus, 

Page  38.     log  sin  21°  37'  =    9.56631  —  10. 

Page  45.     log  cot  36°  53'  -  10. 12473  —  10  =  0.12473. 

Page  37.     log  cos 69°  14'  =    9.54969—10. 

Page  49.     log  tan  45°  59'  =  10.01491  —  10  =  0.01491. 

Page  48.     If  log  cos  =  9.87468  -  10,  angle  =  41°  28'. 

Page  34.     If  log  cot  =  9.39353  -  10,  angle  =  76°  6'. 

If  log  sin  =  9.47760  —  10,  the  nearest  log  sin  in  the  table  is  5.47774  —  10 
(page  36),  and  the  angle  corresponding  to  this  value  is  17°  29'. 

If  log  tan  =  0.76520  =  10.76520  —  10,  the  nearest  log  tan  in  the  table  is 
10.76490  —  10  (page  32),  and  the  angle  corresponding  to  this  value  is  80°  15'. 

26.  If  it  is  desired  to  obtain  the  logarithms  of  the  functions  of 
angles  that  contain  seconds,  or  to  obtain  the  value  of  the  angle  in 
degrees,  minutes,  and  seconds,  from  the  logarithms  of  its  functions, 
interpolation  must  be  employed.  Here  it  must  be  remembered 
that, 

The  difference  between  two  consecutive  angles  in  the  table 
is  60". 

Log  sin  and  log  tan  increase  as  the  angle  increases ;  log  cos  and 
log  cot  diminish  as  the  angle  increases- 


INTRODUCTION.  XV 

Find  log  tan  70°  46'  8". 

Page  37.     log  tan  70°  46'  —  0.45731. 

The  difference  between  the  mantissas  of  log  tan  70°  46'  and  log  tan  70°  47' 
is  41,  and  ^  of  41  =  5. 

As  the  function  is  increasing,  the  5  must  be  added  to  the  figure  in  the  fifth 
place  of  the  mantissa  45731 ;  and 

Therefore  log  tan  70°  46'  8"  =  0.45736. 

Find  log  cos  47°  35'  4". 

Page  48.     log  cos  47°  35'  =  9.82899  -  10. 

The  difference  between  this  mantissa  and  the  mantissas  of  the  next  log  cos 
is  14,  and  -fo  of  14  =  1. 

As  the  function  is  decreasing,  the  1  must  be  subtracted  from  the  figure  in  the 
fifth  place  of  the  mantissa  82899  ;  and 

Therefore  log  cos  47°  35'  4"  =  9.82898  -  10. 

Find  the  angle  for  which  log  sin  =  9.45359  — 10. 

Page  35.     The  mantissa  of  the  nearest  smaller  log  sin  in  the  table  is  45334. 

The  angle  corresponding  to  this  value  is  16°  30'. 

The  difference  between  45334  and  the  given  mantissa,  45359,  is  25. 

The  difference  between  45334  and  the  next  following  mantissa,  45377,  is  43, 
and  ff  of  60"  =  35". 

As  the  function  is  increasing,  the  35"  must  be  added  to  16°  30';  and  the 
required  angle  is  16°  30'  35". 

Find  the  angle  for  which  log  cot  =  0.73478. 

Page  32.     The  mantissa  of  the  nearest  smaller  log  cot  in  the  table  is  73415. 

The  angle  corresponding  to  this  value  is  10°  27'. 

The  difference  between  73415  and  the  given  mantissa  is  63. 

The  difference  between  73415  and  the  next  following  mantissa  is  71,  and  ff 
of  60"  =  53". 

As  the  function  is  decreasing,  the  53"  must  be  subtracted  from  10°  27';  and 
the  required  angle  is  10°  26'  7". 


EXERCISES. 
Find 

1.  log  sin  30°    8'    9".  9.  log  tan  25°  27'  47". 

2.  log  sin  54°  54'  40".  10.  log  cos  56°  11'  57". 

3.  log  cos  43°  32' 31".  11.  log  cot  62°    0'    4" 

4.  log  cos  69°  25'  11".  12.  log  cos  75°  20'  58* 

5.  log  tan  32°    9' 17".  13.  log  tan  33°  27' 13". 

6.  log  tan  50°    2'    2".  14.  log  cot  81°  55' 24". 

7.  log  cot  44°  33' 17".  15.  log  tan  89°  46' 35". 

8.  log  cot  55°    9' 32".  16.  log  tan    1°  25' 56". 


XVI  LOGARITHMS. 

Find  the  angle  A  if 

17.  log  sin  A-    9.70075.  25.  log  cos  A  =  940008: 

18.  logsinJ.=    9.91289.  26.  log  cot  A  -  9.78815. 

19.  log  cos  A—    9.86026.  27.  log  cos  A  =  9.34301. 

20.  log  cos  A-    9.54595.  28.  log  tan  A  =  10. 52288. 

21.  logtan^l  =    9.79840.  29.  log  cot  A  =  965349. 

22.  log  t&nA  =  10.07671.  30.  log  sin  A  =  8.39316. 

23.  log  cot  A  =  10.00675.  31.  log  sin  A  =  8.06678. 

24.  log  cot  A-    9.84266.  32.  logtan^l  =  8.11148. 

27.  If  log  sec  or  log  esc  of  an  angle  is  desired,  it  may  be  found 
from  the  table  by  the  formulas, 

sec  A  =  —  — ;  hence,  log  sec  A  =  colog  cos  A. 

COS  ^L 

esc  A  =  — — 7 ;  hence,  log  esc  A  =  colog  sin  A. 
sin  A 

Page,  31.     log  sec    8°  28'         =  colog  cos   8°  28'         =0.00476. 
Page  42.     log  esc  59°  36'  44"  =  colog  sin  59°  36'  44"  =  0.06418. 

28.  If  a  given  angle  is  between  0°  and  1°,  or  between  89°  and  90°; 
or,  conversely,  if  a  given  log  sin  or  log  cos  does  not  lie  between  the 
limits  8.24186  and  9.99993  in  the  table;    or,  if  a  given  log  tan  or 
log  cot  does  not  lie  between  the  limits  8.24192  and  11.75808  in  the 
table ;  then  pages  21-24  of  Table  III.  must  be  used. 

On  page  21,  log  sin  of  angles  between  0°  and  0°  3',  or  log  cos  of 
the  complementary  angles  between  89°  57'  and  90°,  are  given  to 
every  second;  for  the  angles  between  0°  and  0°  3',  log  taii  =  log  sin, 
and  log  cos  =  0.00000 ;  for  the  angles  between  89°  57'  and  90°, 
log  cot  =  log  cos,  and  log  sin  =  0.00000. 

On  pages  22-24,  log  sin,  log  tan,  and  log  cos  of  angles  between 
0°  and  1°,  or  log  cos,  log  cot,  and  log  sin  of  the  complementary 
angles  between  89°  and  90°,  are  given  to  every  10". 

Whenever  log  tan  or  log  cot  is  not  given,  they  may  be  found  by 
the  formulas, 

log  tan  =  colog  cot.  log  cot  =  colog  tan. 

Conversely,  if  a  given  log  tan  or  log  cot  is  not  contained  in  the 
table,  then  the  colog  must  be  found ;  this  will  be  the  log  cot  or 
log  tan,  as  the  case  may  be,  and  will  be  contained  in  the  table. 

On  pages  25-27  the  logarithms  of  the  functions  of  angles 
between  1°  and  2°,  or  between  88°  and  90°,  are  given  in  the  manner 
employed  on  pages  22-24.  These  pages  should  be  used  if  the  angle 
lies  between  these  limits,  and  if  not  only  degrees  and  minutes,  but 
degrees,  minutes,  and  multiples  of  10"  are  given  or  required. 


INTRODUCTION.  XV11 

When  the  angle  is  between  '0°  and  2°,  or  88°  and  90°,  and  a 
greater  degree  of  accuracy  is  desired  than  that  given  by  the  table, 
interpolation  may  be  employed ;  but  for  these  angles  interpolation 
does  not  always  give  true  results,  and  it  is  better  to  use  Table  IV. 

Find  log  tan  0°  2'  47",  and  log  cos  89°  37'  20". 

Page  21.      log  tan    0°    2'  47"  ~  log  sin  0°  2'  47"  =  6.90829  —  10. 
Page  23.      log  cos  89°  37'  20"  =  7.81911  -  10. 

Find  log  cot  0°  2'  15". 

10  -10 

Page  21.      log  tan  0°  2'  15"     -    6.81591  —  10 
Therefore,  log  cot  0°  2' 15"     =    3.18409 

Find  log  tan  89°  38'  30". 

10  —10 

Page  23.      log  cot  89°  38'  30"  =    7.79617-10 
Therefore,  log  tan  89°  38'  30"  -    2.20383 

Find  the  angle  for  which  log  tan  =  6.92090  — 10. 

Page  21.     The  nearest  log  tan  is  6.92110  —  10. 
The  corresponding  angle  for  which  is  0°  2'  52". 

Find  the  angle  for  which  log  cos  =  7.70240  — 10. 

Page  22.     The  nearest  log  cos  is  7.70261  —  10. 
The  corresponding  angle  for  which  is  89°  42'  40". 

Find  the  angle  for  which  log  cot  =  2.37368. 

This  log  cot  is  not  contained  in  the  table. 
The  colog  eot  =  7.62632  —  10  =  log  tan. 

The  log  tan  in  the  table  nearest  to  this  is  (page  22)  7.62510  —  10,  and  the 
angle  corresponding  to  this  value  of  log  tan  is  0°  14'  30". 

29.  If  an  angle  x  is  between  90°  and  360°,  it  follows,  from 
formulas  established  in  Trigonometry,  that, 

between  90°  and  180°,  between  180°  and  270°, 

log  sin  x  =  log  sin  (180°  —  x),  log  sin  x  =  log  sin  (x  —  180°)n, 

log  cos  x  =  log  cos  (180°  —  x)n,  log  cos  x  =  log  cos  (x  —  180°)n> 

log  tan  x  =  log  tan  (180°  —  »)„,  log  tan  x  =  log  tan  (x  — 180°), 

log  cot  x  =  log  cot  (180°  —  x)n ;  log  cot  x  =  log  cot  (aj  — 180°)  j 

between  270°  and  360°, 
log  sin  x  =  log  sin  (360°  —  aj)B, 
log  cos  x  =  log  cos  (360°  —  x), 
log  tail  x  =  log  tan  (360°  —  x)n, 
log  cot  a:  =  log  cot  (360°  —  «)„. 


XV111  LOGARITHMS. 

The  letter  n  is  placed  (according  to  custom)  after  the  logarithms 
of  those  functions  which  are  negative  in  value. 

The  abovek  formulas  show,  without  further  explanation,  how  to 
find  by  means  of  Table  III.  the  logarithms  of  the  functions  of  any 
angle  between  90°  and  360°. 

Thus,  log  sin  137°  45'  22"  =  log  sin  42°  14'  38"  =  9.82756  —  10. 
log  cos  137°  45'  22"  -  logn  cos  42°  14'  38"  =  9.86940n  -  10. 
log  tan  137°  45'  22"  =  log,,  tan  42°  14'  38"  =  9.95815n  -  10. 
log  cot  137°  45'  22"  =  logn  cot  42°  14'  38"  =  0.04185n. 
log  sin  209°  32'  50"  =  logn  sin  29°  32'  50"  =  9.69297n  —  10. 
log  cos  330°  27'  10"  =  log  cos  29°  32'  50"  -  9.93949  -  10. 

Conversely,  to  a  given  logarithm  of  a  trigonometric  function 
there  correspond  between  0°  and  360°  four  angles,  one  angle  in 
each  quadrant,  and  so  related  that  if  x  denote  the  acute  angle,  the 
other  three  angles  are  180°—  x,  180°  +  x,  and  360°  —  x. 

If  besides  the  given  logarithm  it  is  known  whether  the  function 
is  positive  or  negative,  the  ambiguity  is  confined  to  two  quadrants, 
therefore  to  two  angles. 

Thus,  if  the  log  tan  =  9.47451  —  10,  the  angles  are  16°  36'  17"  in  Quadrant  I. 
and  196°  36'  17"  in  Quadrant  III. ;  but  if  the  log  tan  =  9.47451n—  10,  the  angles 
are  163°  ^3'  43"  in  Quadrant  II.  and  343°  23'  43"  in  Quadrant  IV. 

To  remove  all  ambiguity,  further  conditions  are  required,  or  a 
knowledge  of  the  special  circumstances  connected  with  the  problem 
in  question. 

TABLE   IV. 

30.  This  table  (page  50)  must  be  used  when  great  accuracy  is 
desired  in  working  with  angles  between  0°  and  2°,  or  between  88° 
and  90°. 

The  values  of  S  and  T  are  such  that  when  the  angle  a  is 
expressed  in  seconds, 

S  =  log  sin  a  —  log  a", 
T  =  log  tan  a  —  log  a". 

Hence  follow  the  formulas  given  on  page  50. 

The  values  of  S  and  T  are  printed  with  the  characteristic  10  too 
large,  and  in  using  them  — 10  must  always  be  annexed. 


Find  log  sin  0°  58'  17". 

0°  58'  17"  =  3497" 
log  3497  =  3.54370 

8  =  4.68555  —  10 
log  sin  0°  68'  17"  =  8.22925  -  10 


Find  log  cos  88°  26'  41.2". 

90°  -  88°  26'  41.2"  =  1°  33'  18.8" 

=  5598.8" 
log  5598.8  =  3.74809 

S  =  4.68552  —  10 
log  cos  88°  26'  41.2"  =  8.43361  -  10 


INTRODUCTION.  XIX 


Find  log  tan  0°  52'  47.5", 

0°  52' 47. 5"  =  3167. 5" 

log  3167. 5  =  3. 50072 

T  =  4.68561  -  10 
log  tan  0°  52'  47.5"  =  8.18633  -  10 


Find  log  tan  89°  54'  37.362". 

90°  —  89°  54'  37.362"  =  0°  5'  22.638" 

=  322.638" 
log  322.638  =  2.50871 

T=  4.68558  -10 

log  cot  89°  54'  37.362"  =  7.19429  -  1O 
log  tan  89°  54'  37.362"  =  2.80571 


Find  the  angle,  if  log  sin  =  6.72306  — 10. 

6.72306-10 
S  =  4.68557  -  10 
Subtract,     2.03749          =  log  109.015 

109.015"        =  0°  I'  49.015". 

i 

Find  the  angle  for  which  log  cot  =  1.67604. 

colog  cot  =  8.32396  —  10 
T  =  4.68564  -  10 

Subtract,      3.63832  =  log  4348.3 

4348.3"          =  1°  12'  28.3". 

Find  the  angle  for  which  log  tan  =  1.55407. 

colog  tan  =  8.44593  —  10 
T  =  4.68569-10 
Subtract,      3.76024  =  log  5757.6 

5757.6"  =  1°  35'  57.6", 
and  90°  -  1°  35'  57.6"  =  88°  24'  2.4". 
Therefore,  the  angle  required  is  88°  24'  2. 4". 

TABLE   V. 

31.  This  table  (p.  51),  containing  the  circumferences  and  areas 
of  circles,  does  not  require  explanation. 

TABLE   VI. 

32.  Table  VI.  (pp.  52-69)  contains  the  natural  sines,  cosines, 
tangents,    and    cotangents    of    angles    from   0°   to   90°,    at   inter- 
vals of  1'.      If  greater  accuracy  is  desired  it  may  be  obtained 
by  interpolation. 

NOTE.  In  preparing  the  preceding  explanations,  we  have  made  free  use 
of  the  Logarithmic  Tables  by  F.  G.  Gauss.  For  Table  VI.  we  are  indebted 
to  D.  Carhart. 

TABLE   VII. 

33.  This  table  (pp.  70-75)  gives  the  latitude  and  departure  to 
three  places  of  decimals  for  distances  from  1  to  10,  corresponding 
to  bearings  from  0°  to  90°  at  intervals  of  15'. 


XX 


LOGARITHMS. 


If  the  bearing  does  not  exceed  45°  it  is  found  in  the  Ze/Miand 
column,  and  the  designations  of  the  columns  under  "  Distance " 
are  taken  from  the  top  of  the  page ;  but  if  the  bearing  exceeds 
45°,  it  is  found  in  the  right-h&nd.  column,  and  the  designations 
of  the  columns  under  "  Distance "  are  taken  from  the  bottom  of 
the  page. 

The  method  of  using  the  table  will  be  made  plain  by  the  follow- 
ing examples :  — 

(1)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  N.  35°  15'  E.  6  chains. 

On  p.  75,  left-hand  column,  look  for  35°  15' ;  opposite  this  bearing,  in  the 
vertical  column  headed  "Distance  6,"  are  found  4.900  and  3.463  under  the 
headings  "Latitude"  and  "Departure"  respectively.  Hence,  latitude  or 
northing  =  4.900  chains,  and  departure  or  easting  =  3.463  chains. 

(2)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  S.  87°  W.  2  chains. 

As  the  bearing  exceeds  45°,  we  look  in  the  right-hand  column  of  p.  70,  and 
opposite  87°  in  the  column  marked  ' '  Distance  2  ' '  we  find  (taking  the  designa- 
tions of  the  columns  from  the  bottom  of  the  page)  latitude  =  0.105  chains,  and 
departure  =  1.997  chains.  Hence,  latitude  or  southing  =  0. 105  chains,  and 
departure  or  westing  =  1.997  chains. 

(3)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  1ST.  15°  45'  W.  27.36  chains. 

In  this  case  we  find  the  required  numbers  for  each  figure  of  the  distance 
separately,  arranging  the  work  as  in  the  following  table.  In  practice,  only  the 
last  columns  under  "Latitude  "  and  "  Departure  "  are  written. 


DISTANCE. 

LATITUDE. 

DEPARTURE. 

20       =  2  X  10 
7 
0.3    =3  -=-10 
0.06  =  6  +  100 

1.925  X  10    =  19.25 
6.737 
2.887  -r  10    =0.289 
5.  775  -r  100  =  0.058 

0.543  X  10    =  5.43 
1.90 
0.814  -r  10    =0.081 
1.628  -r  100  =  0.016 

27.36 

26.334 

7.427 

Hence,  latitude  =  26.334  chains,  and  departure  =  7.427  chains. 


TABLE  L 

THE 

COMMON  OE  BEIGGS  LOGAEITHMS 

OF  THE 

NATURAL  NUMBERS 

From  1  to  10000. 

1-100 

N          log 

N          log 

N          log 

N          log 

N          log 

1    0.  00  000 
2    0.  30  103 
3    0.47712 
4    0.  60  206 
5    0.69897 

21     1.32222 
22    1.34242 
23    1.36173 
24    1.38021 
25    1.39794 

41     1.  61  278 

42     1.  62  325 
43     1.  63  347 
44     1.64345 
45     1.65321 

61     1.  78  533 

62    1.  79  239 
63     1.  79  934 
64     1.80618 
65     1.  81  291 

81     1.90849 
82     1.91381 
83     1.  91  908 
84     1.  92  428 
85     1.  92  942 

6    0.77815 

7    0.84510 
8    0.90309 
9    0.  95  424 
10    1.00000 

26    1.  41  497 

27    1.43136 
28    1.44716 
29    1.46240 
30    1.47712 

46     1.  66  276 

47     1.  67  210 
48     1.68124 
49     1.69020 
50     1.69897 

66     1.  81  954 

67     1.  82  607 
68     1.83251 
69    1.83885 
70     1.  84  510 

86     1.93450 
87     1.  93  952 
88     1.94448 
89    1.94939 
90     1.  95  424 

11     1.04139 
12    1.07918 
13    1.11394 
14    1.  14  613 
15    1.  17  609 

31    1.49136 
32    1.50515 
33    1.  51  851 
34    1.53148 
35    1.54407 

51     1.70757 
52     1.  71  600 
53     L  72  428 
54     1.  73  239 
55     1.74036 

71     1.  85  126 

72     1.85733 
73     1.  86  332 
74     1.86923 
75     1.87506 

91     1.  95  904 

92     1.96379 
93     1.  96  848 
94     1.97313 
95     1.  97  772 

16    1.20412 
17    1.23045 
18    1.  25  527 
19    1.27875 
20    1.30103 

36    1.55630 
37    1.56820 
38    1.57978 
39    1.59106 
40    1.60206 

56     1.  74  819 

57     1.75587 
58     1.76343 
59    1.77085 
60     1.  77  815 

76     1.88081 
77     1.88649 
78     1.  89  209 
79    1.89763 
80    1.90309 

96     1.98227 
97     1.98677 
98     1.  99  123 
99     1.99564 
100    2.  00  000 

N          log 

N          log 

N          log 

N          log 

N          log 

1-100 


100-150 


3T 

01234 

56789 

100 

101 
102 
103 
104 

00000  00043  00087  00130  00173 
00432  00475  00518  00561  00604 
00860  00903  00945  00988  01030 
01284  01326  01368  01410  01452 
01703  01745   01787  01828  01870 

00217  00260  00303  00346  00389 
00647  00689  00732  00775,  00817 
01  072  01  115   01  157   01  199  01  242 
01494  01536  01578  01620  01662 
01912  01953   01995  02036  02078 

105 

106 
107 
108 
109 

02119  02160  02202  02243   02284 
02531   02572  02612  02653   02694 
02938  02979  03019  03060  03100 
03342  03383  03423  03463  03503 
03743  03782  03822  03862  03902 

02325   02366  02407  02449  02490 
02735   02776  02816  02857  02898 
03141   03181   03222  03262  03302 
03543   03583  03623   03663   03703 
03941  03981   04021   04060  04100 

no 

111 

112 
113 
114 

04139  04179  04218  04258  04297 
04532  04571   04610  04650  04689 
04922  04961   04999  05038  05077 
05308  05346  05385   05423  05461 
05690  05729  05767  05  805.  05843 

04336    04376  04415   04454  04493 
04727   04766  04805   04844  04883 
05  115   05  154   05  192   05  231   05  269 
05500  05538  05576  05614  05652 
05881  05918  05956  05994  06032 

115 

116 
117 
118 
119 

06070  06108  06145   06183  06221 
06446  06483  06521  06558  06595' 
06819  06856  06893  06930  06967 
07188  07225   07262  07298  07335 
07555   07591  07628  07664  07700 

06258  06296  06333  06371  06408 
06633  06670  06707  06744  06781 
07004  07041   07078  07115   07151 
07372  07408  07445  07482  07518' 
07737  07773  07809  07846  07882 

120 

121 
122 
123 
124 

07918  07954  07990  08027  08063 
08279  08314  08350  08386  08422 
08636  08672  08707  08743  08778 
08991   09026  09061   09096  09132 
09342  09377  09412  09447  09482 

08099  0813^   08171   08207  08243 
08458  08493   08529   08565   08600 
08814  08849  08884  08920  08955 
09167  09202   09237   09272  09307 
09517  09552  09587  09621  09656 

125 

126 
127 
128 
129 

09691  09726  09760  09795   09830 
10037   10072   10106  10140  10175 
10380  10415    10449  10483   10517 
10721   10755   10789  10823   10857 
11059  11093   11126   11160  11193 

09864  09899  09934  09968   10003 
10209   10243    10278   10312   10346 
10551   10585    10619   10653   10687 
10890  10924   10958   10992   11025 
11227   11261    11294   11327   11361 

130 

131 
132 
133 
134 

11394   11428   11461   11494  11528 
11727   11760   11793   11826  11860 
12057   12090   12123    12156   12189 
12385    12418    12450   12483   12516 
12710  12743   12775   12808   12840 

11561    11594   11628   11661    11694 
11893    11926   11959   11992   12024 
12222   12254   12287   12320   12352 
12548   12581    12613    12646   12678 
12  372   12905    12937   12969   13001 

135 

136 
137 
138 
139 

13  033   13  066   13  098   13  130  13  162 
13354   13386   13418   13450  13481 
13672   13704   13735   13767   13799 
13988  14019   14051   14082   14114 
14301   14333   14364   14395   14426 

13  194   13  226   13  258   13  290   13  322 
13513   13545   13577   13609  13640 
13830   13862    13893   13925    13956 
14145    14176   14208   14239   14270 
14457   14489   14520   14551    14582 

140 

141 
142 
143 
144 

14613   14644  14675    14706  14737 
14922   14953   14983    15014   15045 
15229   15259   15290   15320  15351 
15534  15564   15594   15625   15655 
15836   15866   15897   15927   15957 

14768   14799  14829  14860  14891 
15  076   15  106   15  137    15  168   15  198 
15  381   15  412   15  442   15  473   15  503 
15685   15715   15746  15776  15806 
15987   16017   16047  16077   16107 

145 

146 
147 
148 
149 

16137   16167   16197   16227  16256 
16435    16465   16495   16524  16554 
16732   16761   16791   16820  16850 
17026  17056  17085   17114  17143 
17319  17348   17377   17406  17435 

16286  16316  16346  16376  16406 
16584   16613   16643    16673    16702 
16879   16909   16938   16967   16997 
17173   17202  17231   17260  17289 
17464  17493   17522   17551   17580 

150 

17609  17638   17667   176%  17725 

17754  17782  17811   17840  17869 

N 

O          1          2          3         4 

56789 

100-150 


150-200 


N 

01234 

56789 

150 

151 
152 
153 
154 

17609  17638  17667  17696   17725 
17898   17926  17955   17984   18013 
18184  18213   18241   18270  18298 
18469  18498   18526  18554  18583 
18752  18780  18808  18837  18  865. 

17754  17782   17811   17840  17869 
18041   18070  18099   18127   18156 
18327   18355   18384  18412   18441 
18611   18639   18667   18696  18724 
18893   18921   18949  18977   19005 

155 

156 
157 

158 
159 

19033    19061    19089   19117   19145 
19312  19340   19368   19396   19424 
19590  19618   19645    19673   19700 
19866   19893    19921    19948   19976 
20140  20167   20194  20222   20249 

19173    19201    19229   19257   19285 
19451   19479  19507  19535    19562 
19728   19756  19783   19811   19838 
20003   20030  20058  20085   20112 
20276  20303  20330  20358  20385. 

160 

161 
162 
163 
164 

20412  20439  20466  20493  20520 
20683  20710  20737  20763  20790 
20952  20978  21005   21032  21059 
21219  21245   21272   21299  21325 
21484  21511   21537   21564   21590 

20548  20575   20602  20629  20656 
20817  20844  20871   20898  20925 
21085   21112  21139  21165   21192 
21352   21378   21405   21431   21458 
21617  21643   21669  21696  21722 

165 

166 
167 
168 
169 

21748  21775   21801   21827   21854 
22011   22037   22063   22089  22115 
22272  22298   22324  22350  22376 
22531   22557   22583   22608  22634 
22789  22814  22840  22866  22891 

21880  21906  21932   21958   21  985. 
22141   22167   22194  22220  22246 
22401   22427  22453   22479  22505 
22660  22686  22712  22737  22763 
22917  22943   22968  22994  23019 

170 

171 
172 

173 

174 

23  045   23  070  23  096  23  121   23  147 
23300  23325   23350  23376  23401 
23553   23578  23603   23629  23654 
23805   23830  23855   23880  23905 
24055   24080   24105   24130   24155 

23172   23198   23223   23249  23274 
23426  23452  23477  23502   23528 
23679  23704   23729  23754   23779 
23930  23955   23980  24005    24030 
24180  24204  24229  24254   24279 

175 

176 
177 
178 
179 

24304  24329  24353   24378  24403 
24551   24576   24601   24625   24650 
24797   24822   24846   24871   24895 
25042   25066   25091   25115   25139 
25285   25310  25334  25358  25382 

24428  24452   24477   24502   24527 
24674  24699   24724  24748   24773 
24920  24944   24969   24993   25018 
25  164  25  188   25  212   25  237   25  261 
25406  25431   25455   25479  25503 

180 

181 
182 
183 

184 

25527  25551   25575   25600  25624 
25768  25792   25816  25840  25864 
26007   26031   26055   26079  26102 
26245   26269  26293   26316  26340 
26482  26505   26529  26553  26576 

25648  25672  25696  25720  25744 
25888  25912  25935   25959  25983 
26126  26150  26174   26198  26221 
26364  26387   26411   26435   26458 
26600  26623   26647   26670  26694 

185 

186 
187 
188 
189 

26717  26741   26764  26788  26811 
26951   26975   26998   27021   27045 
27184  27207   27231   27254  27277 
27416  27439  27462  27485   27508 
27646  27669  27692  27715   27738 

26834  26858   26881   26905.   26928 
27068  27091   27114  27138  27161 
27300  27323   27346  27370  27393 
27531   27554  27577  27600  27623 
27761   27784  27807  27830  27852 

19O 

191 
192 
193 
194 

27875   27898  27921   27944  27-967 
28103   28126  28149  28171   28194 
28330  28353   28375   28398   28421 
28556  28578  28601   28623   28646 
28780   28803   28825   28847   28870 

27989  28012  28035  28058  28081 
28217   28240  28262   28285    28307 
28443   28466  28488  28511   28533 
28668  28691   28713   28735   28758 
28892   28914   28937   28959  28981 

195 

196 
197 
198 
199 

29003   29026  29048  29070  29092 
29226  29248   29270  29292   29314 
29447   29469  29491    29513   29*535 
29667  29688  29710  29732  29754 
29885   29907   29929  29951   29973 

29115    29137   29159   29181   29203 
29336  29358   29380   29403   29425 
29557   29579  29601   29623   2964*5 
29776  29798  29820  29842   29863 
29994  30016  30038   30060  30081 

2OO 

30103  30125,  30146  30168  30190 

30211  30233  30255  30276  30298 

N 

01234 

56789 

150-200 


200-250 


K 

O          1          2          3         4 

56789 

2OO 

30103  30123  30146  30168  30190 

30211   30233   30255   30276  30298 

201 

30320  30341   30363   30384  30406 

30428  30449  30471   30492  30514 

202 

30535  30557  30578  30600  30621 

30643  30664  30685  30707  30728 

203 

30750  30771  30792  30814  30835 

30856  30878  30899  30920  30942 

204 

30963  30984  31006  31027  31048 

31  069  31  091  31  112  31  133  31  154 

205 

31  175  31  197  31  218  31  239  31  260 

31281   31302  31323  31345  31366 

206 

31387  31408  31429  31450  31471 

31492  31513  31534  31555  31576 

207 

31597  31618  31639  31660  31681 

31  702   31  723   31  744  31  765   31  785 

208 

31806  31827  31848  31869  31890 

31911   31931   31952  31973   31994 

209 

32015   32035   32056  32077  32098 

32118  32139   32160  32181   32201 

21O 

32222  32243   32263   32284  32305 

32325  32346  32366  32387  32408 

211 

32428  32449   32469  32490  32510 

32531   32552  32572  32593   32613 

212 

32634  32654  32675   32695   32715 

32736  32756  32777  32797  32818 

213 

32838  32858  32879   32899  32919 

32940  32960  32980  33001   33021 

214 

33041   33062   33082   33102  33122 

33143  33163  33183  33203  33224 

215 

33244  33264  33284  33304  33325 

33345   33365   33385   33405  33425 

216 

33445   33465   33486  33506  33526 

33546  33566  33586  33606  33626 

217 

33646  33666  33686  33706  33726 

33746  33766   33786  33806  33826 

218 

33846  33866  33885   33905  33925 

33945   33965   33985   34005   34025 

219 

34044  34064  34084  34104  34124 

34143  34163  34183  34203  34223 

22O 

34242  34262  34282   34301   34321 

34341   34361  34380  34400  34420 

221 

34439  34459  34479  34498  34518 

34537  34557  34577  34596  34616 

222 

34635   34655   34674  34694  34713 

34733  34753  34772  34792  34811 

223 

34830  34850  34869  34889  34908 

34928  34947   34967  34986  35005 

224 

35025   35044  35064  35083  35102 

35  122  35  141  35  160  35  180  35  199 

225 

35218  35238  35257  35276  35295 

35315   35334  35353  35372  35392 

226 

35411   35430  35449  35468  35488 

35507  35526  35545  35564  35583 

227 

35603   35622  35641   35660  35679 

35698  35717  35736  35755  35774 

228 

35793  35813  35832  35851  35870 

35889  35908  35927  35946  35965 

229 

35984  36003  36021  36040  36059 

36078  36'  097  36116  36135  36154 

230 

36173  36192  36211   36229  36248 

36267  36286  36305  36324  36342 

231 

36361   36380  36399  36418  36436 

36455   36474  36493   36511   36530 

232 

36549  36568  36586  36605   36624 

36642  36661   36680  36698  36717 

233 

36736  36754  36773  36791  36810 

36829  36847  36866  36884  36903 

234 

36922  36940  36959  36977  36996 

37014  37033  37051  37070  37088 

235 

37107  37125   37144  37162  37181 

37199  37218  37236  37254  37273 

236 

37291  37310  37328  37346  37365 

37383  37401   37420  37438  37457 

237 

37475   37493  37511  37530  37548 

37566  37585   37603  37621  37639 

238 

37  658  37  676  3*  594  37  712  37  731 

37749  37767  37785  37803  37822 

239 

37840  37858  3',    '76  37894  37912 

.37931  37949  37967  37985   38003 

240 

38021  38039  38057  38075  38093 

38112  38130  38148  38166  38184 

241 

38  202  38  220  38  238  38  256  38  274 

38292  38310  38328  38346  38364 

242 

38382  38399  38417  38435  38453 

38471  38489  38507  38525   38543 

243 

38561  38578  38596  38614  38632 

38650  38668  38686  38703   38721 

244 

38739  38757  38775   38792  38810 

38828  38846  38863  38881   38899 

245 

38917  38934  38952  38970  38987 

39005  39023  39041  39058  39076 

246 

39094  39111   39129  39146  39164 

39182   39199  39217   39235   39252 

247 

39270  39287  39  305.   39322  39340 

39358  39375  39393  39410  39428 

248 

39445   39463   39480  39498  39515 

39533  39550  39568  39585   39602 

249 

39620  39637  39655   39672  39690 

39707  39724  39742  39759  39777 

250 

39794  39811  39829  39846  39863 

39881  39898  39915   39933  39950 

N 

O          1          2          3         4 

56789 

200-250 


250-300 


N 

O          1          2          3          4 

56789 

25O 

251 
252 

253 

254 

39794  39811   39829  39846  39863 
39967   39985   40002   40019  40037 
40140  40157   40175   40192   40209 
40312  40329  40346  40364  40381 
40483   40500  40518  40535   40552 

39881   39898  39915   39933   39950 
40054   40071   40088  40106  40123 
40226  40243   40261   40278   40295 
40398  40415   40432   40449  4O466 
40569  40586  40603  40620  40637 

255 

256 

257 
258 
259 

40654   40671   40688  40705   40722 
40824  40841   40858  40875   40892 
40993   41010  41027   41044   41061 
41  162   41  179  41  196  41  212   41  229 
41330  41347  41363   41380  41397 

40739  40756  40773   40790  40807 
40909  40926  40943   40960  40976 
41  078  41  095   41  111   41  128  41  145 
41  246  41  263   41  280  41  296  41  313 
41414  41430  41447   41464   41481 

260 

261 
262 
263 
264 

41497  41514  41531   41547  41564 
41  664   41  681   41  697   41  714  41  731 
41830  41847  41863   41880  41896 
41996  42012  42029  42045   42062 
42  160  42  177   42  193   42  210   42  226 

41581   41597  41614  41631   41647 
41  747  41  764  41  780  41  797   41  814 
41913   41929  41946  41963   41979 
42078  42095   42111   42127   42  H4 
42243   42259  42275   42292  42306 

265 

266 
267 
268 
269 

42325   42341   42357  42374  42390 
42488  42504  42521   42537   42553 
42651   42667  42684  42700  42716 
42813   42830  42846  42862  42878 
42975   42991   43008  43024  43040 

42406  42423   42439  42455   42472 
42570  42586  42602   42619  42635 
42732  42749  42765   42781   42797 
42894  42911   42927   42943   42959 
43  056  43  072  43  088  43  104  43  120 

270 

271 
272 
273 
274 

43  136  43  152  43  169  43  185   43  201 
43297  43313   43329  43345   43361 
43457  43473   43489  43505   43521 
43616  43632  43648  43664  43680 
43  775  43  791   43  807  43  823  43  838 

43217  43233   43249  43265   43281 
43377  43393   43409  43425   43441 
43537  43553  43569  43584  43600 
43696  43712  43727  43743   43759 
43854  43870  43886  43902  43917 

275 

276 

277 
278 
279 

43933  43949  43965  43981   43996 
44091   44107  44122  44138  44154 
44248  44264  44279  44295   44311 
44404   44420  44436  44451   44467 
44560  44576  44592  44607   44623 

44012  44028   44044  44059  44075 
44170  44185   44201   44217   44232 
44326  44342   44358  44373   44389 
44483   44498  44514  44529  44545 
44638  44654  44669  44685   44700 

280 

281 

282 
283 
284 

44716  44731   44747  44762   44778 
44871   44886  44902  44917   44932 
45025   45040  45056  45071   45086 
45  179  45  194  45  209  45  225   45  240 
45332  45347   45362  45378  45393 

44793   44809  44824  44840  44855 
44948  44963   44979  44994  45010 
45  102  45  117  45  133   45  148  45  163 
45255   45271   45286  45301   45317 
45408  45423   45439  45454  45469 

285 

286 

287 
288 
289 

45484  45500  45515   45530  45545 
45637   45652  45667   45682  45697 
45788  45803   45818  45834  45849 
45939  45954  45969  45984  46000 
46090  46105   46120  46135   46150 

45561   45576  45591  45606  45621 
45  712  45  728  45  743   45  758  45  773 
45864  45879  45894  45909  45924 
46015   46030  46045   46060  46075 
46165   46180  46195   46210  46225 

29O 

291 
292 
293 

294 

46240  46255   46270  46285   46300 
46389  46404  46419  46434  46449 
46538  46553   46568   46583   46598 
46687   46702   46716  46731   46746 
46835   46850  46864  46879  46894 

46315   46330  46345   46359  46374 
46464  46479  46494  46509  46523 
46613   46627  46642   46657   46672 
46761   46776  46790  46805   46820 
46909  46923   46938  46953   46967 

295 

296 
297 
298 
299 

46982   46997   47012  47026  47041 
47129  47144  47159  47173   47188 
47276  47290  47305   47319  47334 
47422  47436  47451   47465   47480 
47567  47582  47596  47  61J.   47625 

47056  47070  47085   47100  47114 
47202  47217   47232   47246  47261 
•  47349  47363   47378  47392  47407 
47494  47509  47524  47538  47553 
47640  47654   47669   47683  47698 

3OO 

47712  47727  47741   47756  47770 

47784  47799  47813  47828  47842 

N 

O          1          2          3         4 

56789 

250  -  300 


300-350 


N 

01234 

56789 

3OO 

301 
302 
303 
304 

47712  47727  47741   47756  47770 
47857  47871  47885   47900  47914 
48001   48015   48029  48044  48058 
48144  48159  48173   48187  48202 
48287  48302  48316  48330  48344 

47784  47799  47813  47828  47842 
47929  47943   47958  47972  47986 
48073   48087  48101   48116  48130 
48216  48230  48244  48259  48273 
48359  48373  48387  48401  48416 

3O5 

306 
307 
308 
309 

48430  48444  48458  48473   48487 
48572  48586  48601   48615   48629 
48714  48728  48742  48756  48770 
48855   48869  48883   48897  48911 
48996  49010  49024  49038  49052. 

48501  48515  48530  48544  48558 
48643  48657  48671   48686  48700 
48785   48799  48813   48827   48841 
48926  48940  48954  48968  48982 
49066  49080  49094  49108  49122 

310 

311 
312 
313 
314 

49136  49150  49164   49178  49192 
49276  49290  49304   49318  49332 
49415   49429  49443   49457  49471 
49554  49568  49582  49596  49610 
49693  49707  49721   49734  49748 

49206  49220  49234  49248  49262 
49346  49360  49374  49388  49402 
49485   49499  49513   49527  49541 
49624  49638  49651   49665   49679 
49762  497T&  49790  49803   49817 

315 

316 
317 
318 
319 

49831  49845  49859  49872  49886 
49969  49982   49996   50010  50024 
50106  50120  50133   50147   50161 
50243   50256  50270   50284   50297 
50379  50393  50406  50420  50433 

49900  49914  49927   49941   49955 
50037   50051    50065   50079  50092 
50174   50188   50202   50215   50229 
50311   50325   50338  50352  50365 
50447  50461   50474  50488  50501 

320 

321 

322 
323 
324 

50515   50529  50542   50556  50569 
50651   50664  50678  50691   50705 
50786  50799  50813   50826  50840 
50920  50934  50947  50961   50974 
51  055   51  068  51  081   51  095   51  108 

SO  583   50596  50610  50623   50637 
50718  50732   50745   50759  50772 
50853   50866  50880  50893   50907 
50987   51001    51014   51028  51041 
51121   51135   51148  51162  51175 

325 

326 
327 

328 
329 

51  188   51  202  51  215   51  228   51  242 
51322  51335   51348  51362  51375 
51455   51468  51481   51495   51508 
51587   51601   51614  51627   51640 
51720  51733   51746  51759  51772 

51255   51268  51282   51  295   51308 
51388   51402   51415    51428   51441 
51521    51534   51548   51561   51574 
51654   51667   51680  51693   51706 
51786  51799  51812  51825   51838 

330 

331 
332 
333 
334 

51851   51865   51878  51891   51904 
51983   51996   52009  52022   52035 
52  114   52  127   52  140  52  153   52  166 
52244   52257   52270  52284   52297 
52375   52388   52401   52414   52427 

51917  51930  51943   51957  51970 
52048   52061   52075   52088  52101 
52179   52192   52205   52218  52231 
52310  52323   52336  52349   52362 
52440  52453   52466  52479  52492 

335 

336 
337 
338 
339 

52504  52517  52530  52543   52556 
52634   52647   52660  52673   52686 
52763   52776   52789  52802   52815 
52892   52905    52917   52930   52943 
53020  53033   53046  53058  53071 

52569  52582   52595   52608   52621 
52699   52711   52724   52737   52750 
52827   52840   52853   52866  52879 
52956   52969  52982   52994   53007 
53084  53097  53110  53122  53135 

340 

341 
342 
343 
344 

53  148  53  161   53  173   53  186  53  199 
53275   53288  53301   53314  53326 
53403   53415   53428  53441   53453 
53  529  53  542   53  555   53  567  53  580 
53656  53668  53681   53694  53706 

53212   53224   53237   53250  53263 
53339  53352  53364  53377   53390 
53466  53479   53491   53504  53517 
53593   53605   53618  53631   53643 
53719  53732  53744  53757  53769 

345 

346 
347 
348 
349 

53782  53794  53807   53820  53832 
53908  53920  53933   53945   53958 
54033   54045    54058   54070-54083 
54158   54170  54183   54195   54208 
54283   54295    54307   54320   54332 

53845   53857  53870  53882  53895 
53970  53983   53995    54008  54020 
54095   54108   54120  54133   54145 
54220  54233   54245   54258  54270 
54345   54357    54370   54382  54394 

350 

54407   54419  54432   54444   54456 

54469  54481   54494  54506  54518 

N. 

01234 

56789 

300-350 


350-400 


N 

01234 

56789 

350 

351 
352 
353 
354 

54407   54419  54432   54444   54456 
54531   54543   54555   54568  54580 
54654   54667   54679   54691   54704 
54777  54790  54802  54814  54827 
54900  54913  54  923   54937  54949 

54469  54481   54494   54506  54518 
54593  54605   54617  54630  54642 
54716  54728  54741   54753   54765 
54839  54851   54864  54876  54888 
54962  54974  54986  54998  55011 

355 

356 
357 

358 
359 

55023   55035   55047  55060  55072 
55  145  55  157  55  169  55  182  55  194 
55267  55279  55291   55303  55315 
55388  55400  55413   55425   55437 
55509  55522  55534  55546  55558 

55  084  55  096  55  108  55  121   55  133 
55206  55218  55230  55242   55255 
55328  55340  55352  55364  55376 
55449  55461   55473  55485   55497 
55570  55582  55594  55606  55618 

360 

361 
362 
363 
364 

55630  55642  55654  55666  55678 
55751   55763  55773   55787  55799 
55871   55883   55895  55907  55919 
55991   56003  56015   56027  56038 
56110  56122  56134  56146  56158 

55691   55703   55715   55727  55739 
55811   55823   55835   55847  55859 
55931   55943   55955   55967  55979 
56050  56062  56074  56086  56098 
56170  56182   56194   56205   56217 

365 

366 

367 
368 
369 

56229  56241   56253   56265   56277 
56348  56360  56372  56384  56396 
56467  56478  56490  56502  56514 
56585   56597   56608   56620  56632 
56703   56714  56726  56738  56750 

56289-56301   56312  56324  56336 
56407  56419  56431   56443  56455 
56526  56538  56549  56561   56573 
56644  56656  56667   56679  56691 
56761   56773   56785   56797  56808 

370 

371 
372 
373 
374 

56820  56832  56844  56855   56867 
56937  56949  56961   56972  56984 
57054  57066  57078  57089  57101 
57171   57183   57194  57206  57217 
57287  57299  57310  57322  57334 

56879  56891   56902   56914  56926 
56996  57008  57019  57031   57043 
57113  57124  57136  5714*8  57159 
57229  57241   57252  57264   57276 
57345   57357  57368  57380  57392 

375 

376 
377 
378 
379 

57403   57415   57426  57438  57449 
57519  57530  57542  57553   57565 
57634  57646  57657  57669  57680 
57749  57761   57772  57784  57795 
57864  57875   57887  57898  57910 

57461   57473  57484  57496  57507 
57576  57588  57600  57611*57623 
57692  57703   57715   57726  57738 
57807  57818  57830  57841   57852 
57921  57933   57944  57955   57967 

38O 

381 
382 
383 
384 

57978  57990  58001   58013   58024 
58092  58104  58115   58127  58138 
58206  58218   58229  58240  58252 
58320  58331   58343  58354  58365 
58433   58444  58456  58467  58478 

58035   58047  58058  58070  58081 
58149  58161   58172  58184  58195 
58263   58274   58286   58297   58309 
58377   58388   58399  58410  58422 
58490  58501   58512   58524  58535 

385 

386 
387 
388 
389 

58546  58557  58569  58580  58591 
58659  58670  58681   58692   58704 
58771   58782  58794  58805   58816 
58883   58894  58906  58917  58928 
58995   59006  59017   59028  59040 

58602  58614  58625   58636  58647 
58715   58726  58737  58749  58760 
58  827  58  838  58  850  58  861   58  872 
58939  58950  58961   58973   58984 
59051  59062  59073   59084  59095 

390 

391 
392 
393 

394 

59106  59118  59129  59140  59151 
59218   59229  59240   59251   59262 
59329  59340  59351   59362  59373 
59439  59450  59461   59472   59483 
59550  59561   59572  59583   59594 

59162  59173   59184  59195   59207 
59273   59284   59295   59306  59318 
59384  59395   59406  59417  59428 
59494   59506  59517   59528   59539 
59605   59616  59627  59638  59649 

395 

396 
397 
398 
399 

59660  59671  59682  59693  59704 
59770  59780  59791   59802  59813 
59879  59890  59901   59912   59923 
59988  59999  60010  60021   60032 
60097  60108  60119  60130  60141 

59715  59726  59737  59748  59759 
59824  59835   59846  59857  59868 
59934  59945  59956  59966  59977 
60043   60054   60065   60076  60086 
60152  60163   60173   60184  60195 

4OO 

60206  60217  60228  60239  60249 

60260  60271   60282  60293   60304 

N 

O          1          2          3         4 

5          67          8          9 

350-400 


400-450 


N 

01234 

56789 

400 

401 
402 
403 
404 

60206  60217  60228  60239  60249 
60314  60325   60336  60347  60358 
60423   60433   60444  60455   60466 
60531   60541   60552  60563   60574 
60638  60649  60660  60670  60681 

60260  60271   60  282  60293   60304 
60369  60379  60390  60401   60412 
60477  60487   60498  60509  60520 
60584  60595   60606  60617  60627 
60692   60703   60713   60724  60  735. 

4O5 

406 
407 
408 
409 

60746  60756  60767  60778  60788 
60853   60863   60874  60885   60895 
60  959  60  970  60  981   60  991   61  002 
61066  61077  61087   61098  61109 
61  172  61  183   61  194  61  204  61  215 

60799  60810  60821   60831  60842 
60906  60917   60927   60938  60949 
61013   61023   61034  61045   61055 
61  119  61  130  61  140  61  151   61  162 
61225   61236  61247  61257  61268 

41O 

411 
412 
413 
414 

61278  61289  61300  61310  61321 
61384  61395   61405    61416  61426 
61490  61500  61511    61521   61532 
61595   61606  61616  61627  61637 
61700  61711   61721   61731  61742 

61331   61342  61352   61363   61374 
61437  61448  61458  61469  61479 
61542  61553   61563   61574  61584 
61648  61658  61669  61679  61690 
61752  61763  61773  61784  61794 

415 

416 
417 
418 
419 

61805   61815  61826  61836  61847 
61909  61920  61930  61941   61951 
62014  62024  62034  62045   62055 
62118   62128  62138  62149  62159 
62221   62232  62242  62252  62263 

61857  61868  61878  61888  61899 
61962   61972   61982   61993   62003 
62066  62076  62086  62097  62107 
62170  62180  62190  62201   62211 
62273   62284  62294  62304  62315 

420 

421 
422 

423 
424 

62325   62335  62346  62356  62366 
62428  62439  62449  62459  62469 
62531   62542  62552   62562  62572 
62634  62644  62655   62665   62675 
62737  62747  62757  62767  62778 

62377  62387   62397  62408  62418 
62480  62490  62500  62511   62521 
62583   62593   62603   62613   62624 
62685   62696  62706  62716  62726 
62788  62798  62808  62818  62829 

425 

426 
427 

428 
429 

62839  62849  62859  62870  62880 
'  62941   62951   62961   62972  62982 
63043   63053   63063   63073   63083 
63  144  63  155   63  165   63  175   63  185 
63246  63256  63266  63276  63286 

62890  62900  62910  62921   62931 
62992   63002  63012   63022   63033 
63  094  63  104  63  114  63  124  63  134 
63  195   63  205   63  215   63  225   63  236 
63296  63306  63317  63327  63337 

430 

431 
432 
433 
434 

6334Z  63357  63367  63377  63387 
63448  63458  63468  63478  63488 
63  548  63  558  63  568  63  579  63  589 
63649  63659  63669  63679  63689 
63749  63759  63769  63779  63789 

63397  63407  63417  63428  63438 
63498  63508  63518  63528  63538 
63599   63609  63619  63629  63639 
63  699  63  709  63  719  63  729  63  739 
63799  63809  63819  63829  63839 

435 

436 
437 
438 
439 

63849  63859  63869  63879  63889 
63949  63959  63969  63979  63988 
64048  64058   64068  64078  64088 
64147  64157  64167  64177  64187 
64246  64256  64266  64276  64286 

63899  63909  63919  63929  63939 
63998  64008  64018  64028   64038 
64098  64108  64118   64128  64137 
64197  64207  64217  64227  64237 
64296  64306  64316  64326  64335 

440 

441 
442 

443 
444 

64345  64355  64365   64375   64385 
64444  64454  64464  64473   64483 
64542   64552   64562   64572  64582 
64640  64650  64660  64670  64680 
64738  64748  64758  64768  64777 

64395   64404  64414  64424  64434 
64493   64503   64513   64523   64532 
64591   64601   64611   64621   64631 
64689  64699  64709  64719  64729 
64787  64797  64807  64816  64826 

445 

446 
447 
448 
449 

64836  64846  64856  64865  64875 
64933   64943   64953   64963   64972 
65031  65040  65050  65060  65070 
65128  65137  65147  65157  65167 
65225   65234  65244  65254  65263 

64885   64895   64904  64914  64924 
64982  64992  65002   65011   65021 
65  079  65  089  65  099  65  108  65  118 
65  176  65  186  65  196  65  205   65  215 
65273   65283   65292  65302  65312 

450 

65321  65331  65341  65350  65360 

65369  65379  65389  65398  65408 

N 

01234 

56789 

400-450 


450-500 


N 

O          1          2          3          4 

56789 

450 

65321   65331   65341   65350  65360 

65369  65379  65389  65398  65  406 

451 

65418  65427  65437  65447  65456 

65466  65475  65485   65495  65504 

452 

65514  65523  65533  65543  65552 

65562  65571  65581  65591  65600 

453 

65610  65619  65629  65639  65648 

65658  65667  65677  65686  65696 

454 

65706  65715   65725  65734  65744 

65  753  65  763  65  772  65  782  65  792 

455 

65801   65811   65820  65830  65839 

65849  65858  o5  868  65877  65887 

456 

65896  65906  65916  65925   65935 

65^44  65954  65963  65973   65982 

457 

65992  66001   66011   66020  66030 

66039  66049  66058  66068  66077 

458 

66087   66096  66106  66115   66124 

66134  66143   66153   66162  66172 

459 

66181   66191   66200  66210  66219 

66229  66238  66247   66257   66266 

460 

66276  66285  66295   66304  66314 

66323  66332  66342  66351   66361 

461 

66370  66380  66389  66398  66408 

66417  66427  66436  66445  66455 

462 

66464  66474  66483   66492   66502 

66511   66521   66530  66539  66549 

463 

66558'66567  66577  66586  66596 

66605  66614  66624  66633  66642 

464 

66652  66661  66671   66680  66689 

66699  66708  66717  66727  66736 

465 

66745*66755  66764  66773  66783 

66792  66801   66811   66820  66829 

466 

66839  66848  66857  66867  66876 

66  885   66894  66904  66913  66922 

467 

66932  66941  66950  66960  66969 

66978  66987  66997  67006  67015 

468 

67025   67034   67043   67052  67062 

67071   67080  67089  67099  67108 

469 

67117  67127  67136  67145   67154 

67164  67173  67182  67191   67201 

470 

67210  67219  67228  67237  67247 

67256  67265  67274  67284  67293 

471 

67302  67311  67321   67330  67339 

67348  67357  67367  67376  67385 

472 

67394  67403  67413  67422  67431 

67440  67449  67459  67468  67477 

473 

67486  67495   67504  67514  67523 

67532  67541   67550  67560  67569 

474 

67578  67587  67596  67605   67614 

67624  67633  67642  67651  67660 

^ 

475 

67669  67679  67688  67697  67706 

67715  67724  67733  67742  67752 

476 

67761  '67770  67779  67788  67797 

67806  67815   67825  67834  67843 

477 

67852  67861  67870  67879  67888 

67897  67906  67916  67925   67934 

478 

67943  67952  67961  67970  67979 

67988  67997  68006  68015  68024 

479 

68034  68043  68052  68061   68070 

68079  68088  68097  68106  68115 

48O 

68124  68133  68142  68151   68160 

68169  68178  68187  68196  68205 

481 

68215   68224  68233   68242  68251 

68260  68269  68278  68287  68296 

482 

68305   68314  68323  68332  68341 

68350  68359  68368  68377  68386 

483 

68395   68404  68413   68422   68431 

68440  68449  68458  68467  68476 

484 

68485   68494  68502  68511   68520 

68529  68538  68547  68556  68565 

485 

68574  68583  68592  68601   68610 

68619  68628  68637  68646  68655 

486 

68664  68673  68681  68690  68699 

68708  68717  68726  68735  68744 

487 

68753  68762  68771   68780  68789 

68797  68806  68815  68824  68833 

488 

68842  68851  68860  68869  68878 

68886  68895   68904  68913  68922 

489 

68931  68940  68949  68958  68966 

68975  68984  68993  69002  69011 

490 

69020  69028  69037  69046  69055 

69064  69073  69082  69090  69099 

491 

69108  69117  69126  69135   69144 

69152  69161  69170  69179  69188 

492 

69197  69205   69214   69223   69232 

69241   69249  69258  69267  69276 

493 

69285   69294  69302  69311   69320 

69329  69338  69346  69355  69364 

494 

69373  69381   69390  69399  69408 

69417  69425   69434  69443   69452 

495 

69461  69469  69478  69487  69496 

69504  69513  69522  69531  69539 

496 

69548  69557  69566  69574  69583 

69592  69601   69609  69618  69627 

497 

69636  69644  69653  69662  69671 

69679  69688  69697  69705  69714 

498 

69723  69732  69740  69749  69758 

69767  69775  69784  69793  69801 

499 

69810  69819  69827  69836  69845 

69854  69862  69871   69880  69888 

5OO 

69897  69906  69914  69923  69932 

69  $40  69949  69958  69966  69975 

N 

O          1          2          3         4 

56789 

450-500 


10 


500-550 


N 

01234 

56789 

500 

69897   69906  69914   69923   69932 

69940  69949  69958  69966  69975 

501 

69984  69992   70001    70010   70018 

70027   70036  70044   70053   70062 

502 

70070  70079  70088  70096  70105 

70114   70122   70131    70140   70148 

503 

70157   70165   70174   70183   70191 

70200   70209   70217   70226   70234 

504 

70243   70252   70260   70269   70278 

70286   70295    70303   70312   70321 

5O5 

70329   70338   70346  70355   70364 

70372   70381    70389   70398   70406 

506 

70415    70424   70432   70441    70449 

70458   70467   70475    70484   70492 

507 

70501   70509  70518   70526  70535 

70544   70552-  70561    70569   70578 

508 

70586   70595    70603    70612   70621 

70629   70638   70646   70655    70663 

509 

70672   70680  70689  70697   70706 

70714   70723    70731   70740  70749 

51O 

70757   70766  70774   70783   70791 

70800  70808  70817   70825   70834 

511 

70842   70851   70859   70868   70876 

70885   70893   70902  70910  70919 

512 

70927   Z0935   70944   70952   70961 

70969   70978   70986   70995    71003 

513 

71012   71020   71029   71037   71046 

71054  71063   71071   71079  71088 

514 

71096  71105   71113   71122  71130 

71139  71147  71155   71164  71172 

515 

71181   71',189  71198   71206   71214 

71223   71231    71240   71248   71257 

516 

71265    71273    71282   71290   71299 

71307   71315    71324   71332   71341 

517 

71349  71357   71366  71374   71383 

71391    71399   71408   71416   71425 

518 

71433    71441    71450   71458   71466 

71475    71483   71492   71500   71508 

519 

71517   71.525   71533   71542   71550 

71559  71567   71575   71584  71592 

52O 

71600  71609  71617   71625   71634 

71642  71650  71659  71667   71675 

521 

71  684  71  692   71  700  71  709   71  717 

71  725   71  734   71  742  71  750  71  759 

522 

71  767   71  775   71  784   71  792   71  800 

71809   71817   71825    71834   71842 

523 

71850  71858   71867   71875    71883 

71892   71900   71908   71917   71925 

524 

71933   71941   71950  7J  958   71966 

71  975   71  983   71  991   71  999  72  008 

525 

72016   72024   72032   72041    72049 

72057   72066   72074   72082   72090 

526 

72099   72107   72115    72123    72132 

72140   72148   72156   72165    72173 

527 

72181    72189   72198   72206   72214 

72222   72230   72239   72247   72255 

528 

72263    72272   72280   72288   72296 

72304   72313    72321    72329   72337 

529 

72346  72354  72362   72370   72378 

72387   72395   72403   72411   72419 

530 

72428   72436   72444   72452    72460 

72  469   72  477   72  485  •  72  493    72  501 

531 

72509   72518   72526   72534   72542 

72550   72558   72567   72575    72583 

532 

72591    72599   72607   72616   72624 

72632    72640   72648   72656   72665 

533 

72673   72681    72689   72697    72705 

72713    72722   72730   72738   72746 

534 

72754  72762   72770  72779    72787 

72795    72803    72811    72819   72827 

535 

72835   72843   72852   72860   72868 

72876   72884   72892   72900   72908 

536 

72916   72925    72933    72941    72949 

72957   72965    72973    72981    72989 

537 

72997   73006   73014   73022   73030 

73038   73046  73054   73062   73070 

538 

73078   73086   73094   73102    73111 

73119   73127   73135    7314.3   73151 

539 

73  159  73  167   73  175    73183   73  191 

73199   73207   73215    73223   73231 

54O 

73  239   73  247   73  255    73  263    73  272 

73280   73288   73296   73304   73312 

541 

73320   73328   73336   73344   73352 

73360   73368   73376   73384   73392 

542 

73400   73408   73416   73424   73432 

73440   73448   73456   73464   73472 

543 

73480   73488   73496   73504   73512 

73520   73528   73536   73544   73552 

544 

73560  73568   73576   73584   73592 

73600  73608  73616  73624  73632 

545 

73640  73648  73656   73664   73672 

73  679  73  687   73  695   73  703   73  711 

546 

73719  73727   73735   73743   73751 

73759  73767   73775    73783   73791 

547 

73799  73807  73815   73823   73830 

73838   73846   73854   73862   73870 

548 

73878   73886  73894  73902  73910 

73918   73926  73933    73941    73949 

549 

73957   73965   73973   73981   73989 

73997   74005    74013    74020   74028 

550 

74036  74044  74052  74060  74068 

74076  74084   74092   74099  74107 

N 

01234 

56789 

500-550 


550-600 


ii 


N 

O          1          2          3          4 

56789 

55O 

551 

552 
553 
554 

74036  74044   74052   74060   74068 
74115   74123    74131    74139   74147 
74194   74202   74210   74218   74225 
74273   74280   74288   74296   74304 
74351   74359  74367   74374  74382 

74076  74084  74092  74099  74107 
74155   74162  74170  74178  74186 
74233   74241   74249   74257   74265 
74312   74320  74327  74335   74343 
74390  74398  74406  74414  74421 

555 

556 

557 
558 
559 

74429  74437   74445   74453   74461 
74507  74515   74523   74531   74539 
74586  74593   74601   74609  74617 
74663   74671   74679  74687   74695 
74741   74749  74757   74764  74772 

74468  74476  74484  74492   74500 
74547   74554   74562   74570  74578 
74624   74632   74640   74648   74656 
74  702  74  710  74  718*  74  726  74  733 
74780  74788  74796  74803   74811 

56O 

561 
562 
563 

564 

74819  74827   74834   74842  74850 
74896   74904   74912   74920   74927 
74974  74981   74989  74997  75005 
75051   75059  75066  75074  75082 
75  128  75  136  75  143   75  151   75  159 

74858  74865   74873   74881   74889 
74935   74943   74950  74958  74966 
75012  75020  75028  75035   75043 
75  089  75  097  75  105   75  113   75  120 
75166  75174  75182  75189  75197 

565 

566 
567 
568 
569 

75205   75213   75220  75228  75236 
75282  75289  75297   75305   75312 
75358  75366  75374  75381   75389 
75435   75442   75450  75458  75465 
75511   75519  75526  75534  75542 

75243   75251   75259  75266  75274 
75320  75328  75335   75343   75351 
75397  75404  75412  75420  75427 
75473   75481   75488  75496  75504 
75549  75557   75565   75572  75580 

57O 

571 
572 
573 
574 

75587   75595   75603   75610  75618 
75664  75671   75679  75686  75694 
75  740  75  747  75  755   75  762  75  770 
75815   75823   75831   75838  75846 
75891   75899  75906  75914  75921 

75626  75633   75641   75648  75656 
75702  75709  75717   75724  75732 
75778  75785   75793   75800  75808 
75853   75861   75868  75876  75884 
75929  75937   75944  75952  75959 

575 

576 

577 
578 
579 

75967  75974  75982   75989  75997 
76042  76050  76057   76065   76072 
76118  76125   76133   76140  76148 
76193   76200   76208   76215    76223 
76268  76275   76283   76290  76298 

76005   76012  76020  76027  76035 
76080  76087   76095   76103   76110 
76155   76163   76170  76178  76185 
76230  76238  76245   76253   76260 
76305   76313   76320  76328  76335 

580 

581 

582 
583 
584 

76343   76350  76358  76365   76373 
76418  76425   76433   76440  76448 
76492  76500  76507   76515   76522 
76567  76574  76582  76589  76597 
76641   76649  76656  76664  76*671 

76380  76388  76395   76403   76410 
76455   76462   76470  76477  76485 
76530  76537  76545   76552  76559 
76604  76612  76619  76626  76634 
76678  76686  76693   76701   76708 

585 

586 
587 
588 
589 

76716  76723   76730  76738  76745 
76790  76797  76805   76  812  76819 
76864  76871   76879  76886  76893 
76938  76945   76953   76960  76967 
77012  77019  77026  77034  77041 

76753  76760  76768  76775   76782 
76827  76834   76842  76849  76856 
76901    76908  76916  76923   76930 
76975   76982   76989  76997   77004 
77048  77056  77063   77070  77078 

59O 

591 
592 
593 
594 

77085   77093   77100  77107   77115 
77159  77166  77173   77181   77188 
77232   77240   77247    77254   77262 
77305   77313   77320  77327   77335 
77379  77386  77393   77401   77408 

.77122  77129  77137  77144  77151 
77195   77203   77210  77217  77225 
77269  77276  77283   77291   77298 
77342  77349  77357   77364   77371 
77415   77422  77430  77437   77444 

595 

596 
597 
598 
599 

77452  77459  77466  77474  77481 
77525   77532  77539  77546  77554 
77597  77605   77612  77619  77627 
77670  77677   77685   77692   77699 
77743  77750  77757  77764  77772 

77488  77495   77503  77510  77517 
77561   77568  77576  77583   77590 
77634  77641   77648  77656  77663 
77706  77714  77721   77728  77735 
77779  77786  77793   77801   77808 

600 

77815   77822  77830  77837   77844 

77851   77859  77866  77873   77880 

N 

©          1          2          3         4 

56789 

550-600 


12 


600-650 


N 

01234 

56789 

6OO 

601 
602 
603 
604 

77815   77822  77830  77837  77844 
77887   77895   77902  77909  77916 
77960  77967  77974  77981   77988 
78032  78039  78046  78053   78061 
78104  78111   78118   78125   78132 

77851   77859  77866  77873   77880 
77924  77931   77938   77945   77952 
77996  78003   78010  78017   78025 
78068  78075   78082  78089  78097 
78140  78147   78154  78161   78168 

605 

606 
607 
608 
609 

78176  78183   78190  78197   78204 
78247   78254  78262  78269  78276 
78  319  78  326  78  333   78  340  78  347 
7839G  78398  78405   78412  78419 
78462  78469  78476  78483   78490 

78211   78219  78226  78233   78240 
78283   78290  78297   78305   78312 
78355   78362  78369  78376  78383 
78426  78433   78440  78447   78455 
78497  78504   78512  78519  78526 

61O 

611 
612 
613 
614 

78533   78540  78547   78554  78561 
78604  78611   78618   78625   78633 
78675   78682  78689   78696  78704 
78746  78753   78760   78767  78774 
78817   78824  78831   78838  78845 

78569  78576  78583   78590   78597 
78640  78647  78654  78661   78668 
78711   78718  78725   78732  78739 
78781   78789  78796  78803   78810 
78852  78859  78866  78873   78880 

615 

616 
617 
618 
619 

78888  78895   78902   78909  78916 
78958  78965   78972  78979  78986 
79029  79036  79043   79050  79057 
79099  79106  79113   79120  79127 
79169  79176  79183   79190  79197 

78923   78930  78937   78944  78951 
78993   79000  79007   79014  79021 
79064  79071   79078  79085   79092 
79134  79141   79148  79155   79162 
79204  79211   79218  79225   79232 

62O 

621 
622 
623 
624 

79239  79246  79253   79260  79267 
79309  79316  79323   79330  79337 
79379  79386  79393   79400  79407 
79449  79456  79463   79470  79477 
79518  79525   79532   79539  79546 

79274  79281   79288  79295   79302 
79344  79351   79358  79365   79372 
79414  79421   79428  79435   79442 
79484  79491   79498  79505   79511 
79553   79560  79567  79574  79581 

625 

626 
627 
628 
629 

79588  79595   79602  79609  79616 
79657  79664   79671   79678  79685 
79727  79734  79741   79748  79754 
79796  79803   79810  79817  79824 
79865   79872  79879  79886   79893 

79  623   79  630  79  637   79  644  79  650 
79692  79699  79706  79713   79720 
79761   79768  79775   79782  79789 
79831   79837   79844  79851   79858 
79900  79906  79913   79920  79927 

630 

631 
632 
633 
634 

79934  79941   79948  79955   79962 
80003  80010  80017  80024  80030 
80072  80079  80085  80092  80099 
80140  80147  80154  80161  80168 
80209  80216  80223  80229   80236 

79969  79975   79982  79989  79996 
80037  80044  80051  80058  80065 
80106  80113  80120  80127  80134 
80175  80182  80188  80195   80202 
80243   80250  80257  80264  80271 

635 

636 
637 
638 
639 

80277  80284  80291   80298  80305 
80346  80353  80359  80366  80373 
80414  80421   80428  80434  80441 
80482  80489  80496  80502  80509 
80550  80557  80564  80570  80577 

80312  80318  80325  80332  80339 
80380  80387  80393  80400  80407 
80448  80455   80462  80468  80475 
80516  80523  80530  80536  80543 
80584  80591   80598  80604  80611 

640 

641 
642 
643 
644 

80618  80625   80632  80638  80645 
80686  80693  80699  80706  80713 
80754  80760  80767  80774  80781 
80821  80828  80835   80841  80848 
80889  80895   80902  80909  80916 

80652  80659  80665   80672  80679 
80720  80726  80733  80740  80747 
80787  80794    80801   80808  80814 
80855   80862  80868  80875   80882 
80922  80929  80936  80943   80949 

645 

646 
647 
648 
649 

80956  80963  80969  80976  80983 
81023  81030  81037  81043  81050 
81  090  81  097  81  104  81  111   81  117 
81158  81164  81171  81178  81184 
81224  81231  81238  81245  81251 

80990  80996  81003  81010  81017 
81057  81064  81070  81077  81084 
81  124  81  131   81  137   81  144  81  151 
81191   81198  81204  81211   81218 
81258  81265   81271   81278  81285 

65O 

81291  81298  81305  81311  81318 

81325  81331  81338  81345  81351 

N 

01234 

56789 

600-650 


650-700 


13 


N 

O          12          3          4 

5          6          7          8          9 

65O 

651 
652 
653 
654 

81291   81298  81305   81311   81318 
81358  81365   81371  81378  81385 
81425   81431   81438   81445   81451 
81491   81498  81505   81511   81518 
81558  81564  81571   81578  81584 

81325   81331  81338  81345    81351 
81391   81398   81405   81411    81418 
81458  81465   81471   81478  81485 
81525   81531  81538   81544  81551 
81591  81598  81604  81611  81617 

655 

656 

657 
658 
659 

81624   81631    81637   81644  81651 
81  690  81  697   81  704  81  710  81  717 
81  757  81  763   81  770  81  776  81  783 
81823   81829   81836   81842  81849 
81889  81895   81902   81908  81915 

81657  81664  81671   81677  81684 
81723   81730  81737  81743   81750 
81790  81796  81803   81809  81816 
81856  81862   81869   81875   81882 
81921   81928  81935   81941   81948 

660 

661 
662 
663 
664 

81954  81961   81968  81974  81981 
82020  82027   82033    82040  82046 
82086  82092   82099   82105   82112 
82151   82158   82164   82171   82178 
82217   82223    82230   82236  82243 

81987  81994  82000  82007  82014 
82053   82060  82066   82073   82079 
82119   82125   82132   82138   82145 
82184  82191   82197   82204  82210 
82249  82256  82263   82269  82276 

665 

666 
667 
668 
669 

82282   82289   82295    82302  82308 
82347   82354  82360  82367   82373 
82413   82419   82426  82432  82439 
82478  82484   82491   82497   82504 
82543   82549   82556  82562   82569 

82315   82321   82328   82334  82341 
82380  82387   82393   82400  82406 
82445   82452  82458   82465    82471 
82510  82517   82523   82530  82536 
82575   82582  82588  82595   82601 

670 

671 
672 
673 
674 

82607  82614  82620  82627  82633 
82672  82679   82685   82692   82698 
82737   82743   82750  82756  82763 
82802   82808   82814  82821   82827 
82866  82872  82879  82885   82892 

82640  82646  82653  82659  82666 
82705   82711   82718   82724  82730 
82769  82776  82782  82789  82795 
82834  82840  82847   82853   82860 
.  82898  82905   8291]    82918  82924 

675 

676 

677 
678 
679 

82930  82937  82943   82950  82956 
82995   83001   83008  83014  83020 
83059  83065   83072  83078  83085 
83  123   83  129  83  136  83  142   83  149 
83187  83193   83200  83206  83213 

82963   82969  82975  82982  82988 
83027  83033   83040  83046  83052 
83091   83097  83104  83110  83117 
83  155   83  161   83  168  83  174  83  181 
83219  83225   83232  83238  83245 

68O 

681 
682 
683 
684 

83251   83257  83264  83270  83276 
83315   83321   83327  83334  83340 
83378  83385   83391   83398  83404 
83442   83448  83455   83461   83467 
83506  83512  83518  83525   83531 

83283   83289  83296  83302  83308 
83347   83353   83359   83366  83372 
83410  83417   83423   83429  83436 
83474  83480  83487  83493   83499 
83537  83544  83550  83556  83563 

685 

686 
687 
688 
689 

83569  83575   83582  83588  83594 
83632  83639  83645   83651   83658 
836%  83702   83708  83715   83721 
83  759  83  765   83  771   83  778  83  784 
83822  83828,83835   83841   83847 

83601   83607   83613   83620  83626 
83664  83670  83677   83683   83689 
83727  83734  83740  83746  83753 
83790  83797  83803   83809  83816 
83853   83860  83866  83872  83879 

690 

691 
692 
693 
694 

83885   83891   83897  83904  83910 
83948  83954  83960  83967  83973 
84011   84017   84023   84029  84036 
84073   84080  84086  84092  84098 
84136  84142   84148   84155   84161 

83  916  83  923  83  929  83  935   83  942 
83979  83985   83992   83998  84004 
84042  84048  84055   84061   84067 
84105   84111   84117   84123   84130 
84167   84173   84180  84186  84192 

695 

696 
697 
698 
699 

84198   84205   84211   84217   84223 
84261   84267  84273   84280  84286 
84323   84330   84336  84342   84348 
84386  84392   84398   84404  84410 
84448  84454  84460  84466  84473 

84230  84236  84242  84248   84255 
84292  84298  84305   84311   84317 
84354   84361   84367   84373   84379 
84417   84423   84429  84435   84442 
84479  84485    84491   84497  84504 

70O 

84510  84516  84522  84528  84535 

84541   84547   84553   84559  84566 

N 

O          1          2          3          4 

56789 

650-700 


14 


700-750 


N 

01234 

56789 

TOO 

701 
702 
703 
704 

84510  84516  84522  84528  84535 
84572  84578  84584  84590  84597 
84634  84640  84646  84652  84  658 
84696  84702  84708  84714  84720 
84757  84763  84770  84776  84782 

84541   84547  84553  84559  84566 
84603   84609  84615   84621    84628 
84665   84671   84677   84683   84689 
84726  84733   84739   84745   84751 
84788  84794  84800  84807  84813 

7O5 

706 
707 
708 
709 

84819  84825  84831  84837  84844 
84880  84887  84893  84899  84905 
84942  84948  84954  84960  84967 
85003  85009  85016  85022  85028 
8506^  85071  85077  85083  85089 

84850  84856  84862  84868  84874 
84911   84917   84924   84930  84936 
84973   84979   84985    84991   84997 
85034  85040   85046   85052   85058 
85  095   85  101   85  107  85  114  85  120 

710 

711 
712 
713 
714 

85126  85132  85138  85144  85150 
85  187  85  193  85  199  85  205  85  211 
85248  85254  85260  85266  85272 
85309  85315  85321  85327  85333 
85370  85376  85382  85388  85394 

85156  85163  85169  85175   85181 
85217   85224   85230  85236  85242 
85278  85285   85291   85  297  85  303 
85339  85345   85352  85358  85364 
85400  85406  85412  85418  85425 

715 

716 
717 
718 
719 

85431  85437  85443  85449  85455 
85491  85497  85503  85509  85516 
85552  85558  85564  85570  85576 
85612  85618  85625  85631  85637 
85673  85679  85685  85691  85697 

85461   85467   85473  85479  85485 
85522  85528  85534  85540  85546 
85582  85588  85594  85600  85606 
85643  85649  85655   85661  85667 
85703  85709  85715   85721   85727 

720 

721 
722 
723 

724 

85733  85739  85745  85751  85757 
85794  85800  85806  85812  85818 
85854  85860  85866  85872  85878 
85914  85920  85926  85932  85938 
85974  85980  85986  85992  85998 

85763  85769  85775   85781   85788 
85824  85830  85836  85842   85848 
85884  85890  85896  85902  85908 
85944  85950  85956  85962   85968 
86004  86010  86016  86022  86028 

725 

726 

727 
728 
729 

86034  86040  86046  86052  86058 
86094  86100  86106  86112  86118 
86153  86159  86165  86171  86177 
86213  86219  86225  86231  86237 
86273  86279  86285  86291  86297 

86064  86  070  '86  076  86082  86088 
86124  86130  86136  86141   86147 
86183   86189   86195   86201   86207 
86243   86249   86255    86261   86267 
86303  86308  86314  86320  86326 

730 

731 
732 
733 
734 

86332  86338  86344  86350  86356 
86392  86398  86404  86410  86415 
86451  86457  86463  86469  86475 
86510  86516  86522  86528  86534 
86570  86576  86581  86587  86593 

86362  86368  86374  86380  86386 
86421   86427   86433   86439   86445 
86481    86487   86493   86499  86504 
86540  86546  86552   86558  86564 
86599  86605   86611   86617  86623 

735 

736 

737 
738 
739 

86629  86635  86641  86646  86652 
86688  86694  86700  86705  86711 
86747  86753  86759  86764  86770 
86806  86812  86817  86823  86829 
86864  86870  86876  86882  86888 

86658  86664  86670  86676  86682 
86  717  .86  723   86  729  86  735   86  741 
86776  86782   86788   86794   86800 
86835   86841   86847   86853   86859 
86894  86900  86906  86911   86917 

74O 

741 
742 
743 
744 

86923  86929  86935  86941  8694,7 
86982  86988  86994  86999  87005 
87040  87046  87052  87058  87064 
87099  87105  87111  87116  87122 
87157  87163  87169  87175  87181 

86953   86958  86964  86970  86976 
87011  §7017  87023   87029  87035 
87070  87075   87081   87087  87093 
87128  87134  87140  87146  87151 
87186  87192  87'198  87204  87210 

745 

746 

747 
748 
749 

87216  87221  87227  87233  87239 
87274  87280  87286  87291  87297 
87332  87338  87344  87349  87355 
87390  87396  87402  87408  87413 
87448  87454  87460  87466  87471 

87245   87251   87256  87262  87268 
87303  87309  87315   87320  87326 
87361   87367  87373   87379  87384 
87419  87425   87431   87437   87442 
87477  87483   87489  87495   87500 

75O 

87506  87512  87518  87523  87529 

87535  87541   87547   87552  87558 

N 

O          1          2          3          4 

56789 

700-750 


750-800 


15 


N 

O          1          2          3          4 

56789 

75O 

751 
752 
753 
754 

87506  87512  87518  87523   87529 
87564  87570  87576  87581   87587 
87622  87628  87633   87639  87645 
87679  87685   87691   87697  87703 
87737  87743   87749  87754  87760 

87535   87541  87547  87552  87558 
87593   87599  87604  87610  87616 
87651   87656  87662   87668  87674 
87708  87714  87720  87726  87731 
87766  87772  87777  87783  87789 

755 

756 

757 
758 
759 

87795   87800  87806  87812  87818 
87852  87858   87864   87869  87875 
87910  87915   87921   87927   87933 
87967  87973  87978  87984  87990 
88024  88030  88036  88041   88047 

87823  87829  87835   87841   87846 
87881   87887  87892  87898  87904 
87938  87944  87950  87955   87961 
87996  88001   88007  88013  88018 
'88053  88058  88064  88070  88076 

76O 

761 
762 
763 
764 

88081   88087   88093   88098  88104 
88138  88144  88150  88156  88161 
88195   88201   88207   88213   88218 
88252  88258  88264  88270  88275 
88309  88315   88321   88326  88332 

88110  88116  88121   88127  88133 
88167  88173  88178  88184  88190 
88224  88230  88235   88241   88247 
88281   88287  88292   88298  88304 
88338  88343  88349  88355   88360 

765 

766 
767 
76S 
769 

88366  88372  88377  88383  88389 
88423   88429  88434  88440  88446 
•  88  480  88  485   88  491   88  497   88  502 
88536  88542   88547   88553   88559 
88593   88598  88604  88610  88615 

88395   88400  88406  88412  88417 
88451  88457  88463  88468  88474 
88508  88513  88519  88525   88530 
88564  88570  88576  88581   88587 
88621   88627   88632  88638  88643 

77O 

771 
772 
773 
774 

88649  88655   88660  88666  88672 
88705   88711   88717   88722   88728 
88762  88767  88773  88779  88784 
88818  88824   88829  88  835  '88840 
88874  88880  88885   88891   88897 

88677  88683   88689  88694  88700 
88734  88739  88  745_  88750  88756 
88790  88795   88801   88807   88812 
88846  88852   88857   88863   88868 
88902  88908  88913   88919  88925 

775 

776 

777 
778 
779 

88930  88936  88941   88947  88953 
88986  88992   88997  89003   89009 
89042   89048  89053   89059  89064 
89098   89104   89109  89115   89120 
89154   89159  89165   89170  89176 

88958  88964  88969  88975   88981 
89014  89020  89025   89031   89037 
89070  89076  89081   89087   89092 
89126  89131   89137   89143   89148 
89182  89187  89193  89198  89204 

78O 

781 
782 
783 
784' 

89209   89215   89221   89226  89  232 
89265    89271   89276  89282  89287 
89321   89326   89332   89337  89343 
89376  89382  89387  89393   89398 
89432   89437   89443   89448  89454 

89237   89243   89248  89254   89260 
89293   89298  89304  89310  89315 
89348  89354  89360  89365   89371 
89404  89409  89415   89421   89426 
89459  89465   89470  89476  89481 

785 

786 

787 
788 
789 

89487   89492   89498  89504   89509 
89542   89548   89553   89559   89564 
89597   89603   89609  89614  89620 
89653   89658   89664  89669  89675 
89708  89713   89719  89724  89730 

89515   89520  89526  89531   89537 
89570  89575   89581   89586  89592 
S962;>   89631   89636  89642  89647 
89680  89686  89691    89697   89702 
89735   89741   89746  89752  89757 

790 

791 
792 
793 
794 

89763   89768   89774   89779   89785 
89818   89823   89829   89834  89840 
89873   89878   89883    89889  89894 
89927   89933   89938   89944   89949 
89982  89988  89993   89998  90004 

89790  89796  89801   89807  89812 
89845   89851   89856  89862  89867 
89900  89905   89911   89916  89922 
89955   89960  89966  89971    89977 
90009  90015   90020  90026  90031 

795 

796 

797 
798 
799 

90037   90042  90048  90053   90059 
90091   90097  90102   90108  90113 
90146  90151   90157  90162   90168 
90200  90206  90211   90217  90222 
90255   90260  90266  90271  90276 

90064  90069  90075   90080  90086 
90119  90124  90129  90135   90140 
90173   90179  90184  90189  90195 
90227  90233   90238  90244  90249 
90282  90287   90293   90298  90304 

8OO 

90309  90314  90320  90325  90331 

90336  90342  90347  90352  90358 

N 

01234 

56789 

750-800 


16 


800-850 


N 

01234 

56789 

8OO 

801 

802 
803 
804 

90309  90314  90320  90325   90331 
90363   90369  90374  90380  90385 
90417   90423   90428  90434  90439 
90472   90477  90482   90488  90493 
90526  90531   90536  90542  90547 

90336  90342  90347   90352  90358 
90390  90396  90401   90407  90412 
90445   90450  90455   90461   90466 
90499  90504  90509  90515   90520 
90553  90558  90563   90569  90574 

8O5 

806 
807 
808 
809 

90580  90585   90590  90596  90601 
90634  90639  90644  90650  90655 
90687   90693   90698  90703   90709 
90  741   90  747   90  752   90  757-  90  763 
90795   90800  90806  90811   90816 

90607  90612   90617   90623   90628 
90660  90666  90671   90677   90.682 
90714  90720  90725   90730  90736 
90768  90773   90779   90784  90789 
90822   90827   90832   90838  90843 

81O 

811 
812 
813 
814 

90849   90854   90859  90865   90870 
90902   90907   90913   90918  90924 
90956  90961   90966  90972   90977 
91009   91014   91020  91025   91030 
91062   91068   91073   91078  91084 

90875   90881   90886  90891   90897 
90929  90934   90940  90945   90950 
90  982   90  988  90  993   90  998  91  004 
91036  91041   91046  91052  91057 
91089  91094  91100  91105   91110 

815 

816 

817 
818 
819 

91  116  91  121   91  126  91  132  91  137 
91  169  91  174   91  180   91  185   91  190 
91  222  91  228  91  233   91  238  91  243 
91  275   91  281   91  286  91  291   91  297 
91328   91334  91339  91344   91350 

91  142  91  148  91  153   91  158  91  164 
91  196  91  201   91  206  91  212  91  217 
91249  91254   91259  91265  91270 
91302  91307   91312  91318  91323 
91355   91360  91365   91371   91376 

82O 

821 
822 
823 
824 

91381   91387  91392  91397  91403 
91434  91440  91445   91450  91455 
91487   91492  91498   91503   91508 
91540  91545   91551   91556  91561 
91593   91598  91603  91609  91614 

91408   91413   91418  91424  91429 
91461   91466  91471   91477   91482 
91514  91519  91524  91529  91535 
91  566  91  572   91  577  91  582   91  587 
91619  91624  91630  91635   91640 

825 

826 
827 
828 
829 

91645   91651   91656  91661   91666 
91  698  91  703   91  709  91  714  91  719 
91751   91756  91761   91766  91772 
91803   91808  91814  91819  91824 
91855   91861   91866  91871   91876 

91672  91677   91682  91687   91693 
91  724   91  730  91  735   91  740  91  745 
91777  91782  91787  91793  91798 
91  829  91  834  91  840  91  845   91  850 
91882  91887  91892  91897  91903 

830 

831 
832 
833 
834 

91908  91913   91918  91924  91929 
91960  91965    91971   91976  91981 
92012  92018  92023   92028  92033 
92065   92070  92075   92080  92085 
92117   92122   92127  92132    92137 

91934   91939  91944  91950  91955 
91986  91991   91997  92002  92007 
92038   92044   92049  92054  92059 
92091    92096  92101   92106  92111 
92143   92148  92153   92158  92163 

835 

836 
837 
838 
839 

92169  92174   92179  92184   92189 
92221   92226   92231   92236   92241 
92273   92278   92283   92288  92293 
92324  92330  92335   92340   92345 
92376  92381    92387   92392  92397 

92  195   92  200  92  205   92  210  92  215 
92247   92252  92257   92262   92267 
92298  92304  92309  92314  92319 
92350  92355   92361   92366  92371 
92402  92407   92412  92418  92423 

840 

841 
842 
843 
844 

92428   92433    92438   92443   92449 
92480   92485    92490  92495   92500 
92531    92536  92542   92547   92552 
92583   92588   92593   92598  92603 
92634    92639    92645   92650  92655 

92454  92459  92464   92469  92474 
92505   92511   92516  92521   92526 
92557   92562   92567   92572  92578 
92609  92614  92619  92624  92629 
92660  92665   92670  92675   92681 

845 

846 
847 
848 
849 

92686   92691    92696  92701   92706 
92737    92742   92747  92752  92758 
92788   92793   92799  92804  92809 
92840   92845   92850  92855   92860 
92891    92896  92901   92906  92911 

92  711   92  716   92  722*  92  727  92  732 
92763   92768  92773    92778   92783 
92814  92819   92824   92829  92834 
92865   92870  92875    92881   92886 
92916  92921   92927   92932   92937 

850 

92942   92947  92952  92957  92962 

92967   92973   92978   92983   92988 

N 

01234 

56789 

800-850 


850-900 


17 


N 

O          1          2          3          4 

56789 

85O 

851 
852 
853 
854 

92942  92947  92952  92957   92962 
92993   92998   93003   93008  93013 
93044   93049  93054  93059  93064 
93  095   93  100  93  105   93  110  93  115 
93146  93151   93156  93161  93166 

92967   92973   92978  92983   92988 
93018  93024  93029  93034  93039 
93069  93075   93080  93085   93090 
93  120  93  125   93  131   93  136  93  141    . 
93  171   93  176  93  181   93  186  93  192 

855 

856 
857 
858 
859 

93197   93202   93207   93212  93217 
93247  93252  93258  93263   93268 
93298  93303  93308  93313  93318 
93349  93354  93359  93364   93369 
93399  93404  93409  93414   93420 

93222  93227   93232  93237  93242 
93273   93278  93283   93288  93293 
93323   93328  93334  93339  93344 
93374  93379  93384  93389  93394 
93425   93430  93435   93440  93445 

86O 

861 
862 
863 
864 

93450  93455   93460  93465   93'470 
93500  93505   93510  93515   93520 
93551  93556  93561   93566  93571 
93601   93606  93611   93616  93621 
93651  93656  93661  93666  93671 

93475   93480  93485   93490  93495 
93526  93531   93536  93541   93546 
93576  93581   93586  93591   93596 
93626  93631   93636  93641   93646 
93676  93682  93687   93692   93697 

865 

866 
867 
868 
869 

93702  93707  93712  93717  93722 
93752  93757  93762  93767  93772 
93802  93807  93812  93817   93822 
93852  93857  93862  93867  93872 
93902   93907   93912   93917   93922 

93727  93732   93737   93742   93747 
93777  93782  93787  93792  93797 
93827  93832   93837   93842   93847 
93877   93882   93887   93892   93897 
93927   93932   93937   93942  93947 

870 

871 
872 
873 
874 

93952  93957  93962  93967  93972 
94002  94007   94012  94017   94022 
94052   94057  94062  94067  94072 
94101   94106  94111   94116  94121 
94151   94156  94161   94166  94171 

93977  93982  93987  93992  93997 
94027  94032  94037   94042   94047 
94077   94082  94086  94091   94096 
94126  94131   94136  94141   94146 
94176  94181   94186  94191   94196 

875 

876 
877 
878 
879 

94201   94206  94211   94216  94221 
94250  94255    94260  94265   94270 
94300  94305   94310  94315   94320 
94349  94354  94359  94364  94369 
94399  94404   94409  94414  94419 

94226  94231   94236  94240  94245 
94275   94280  94285   94290  94295 
94325   94330  94335   94340  94345 
94374  94379  94384  94389  94394 
94424  94429  94433   94438   94443 

880 

881 
882 
883 
884 

94448  94453   94458  94463   94468 
94498  94503   94507   94512  94517 
94547  94552   94557   94562   94567 
94596  94601   94606  94611   94616 
94645   94650  94655   94660  94  6&5 

94473   94478  94483   94488   94493 
94522  94527   94532   94537  94542 
94571   94576  94581   94586  94591 
94621   94626  94630  94635   94640 
94670  94675   94680  94685   94689 

885 

886 
887 
888 
889 

94694  94699  94704  94709  94714 
94743  94748  94753  94758  94763 
94792   94797   94802   94807   94812 
94841   94846  94851   94856  94861 
94890  94895   94900  94905   94910 

94719  94724   94729  94734   94738 
94768  94773   94778  94783   94787 
94817  94822   94827   94832  94836 
94866  94871   94876  94880  94885 
94915   94919  94924   94929  94934 

89O 

891 
892 
893 
894 

94939  94944  94949  94954  94959 
94988  94993  94998  95002  95007 
95036  95041   95046  95051   95056 
95  085   95  090  95  095   95  100  95  105 
95  134  95  139  95  143   95  148  95  153 

94963  94968  94973  94978  94983 
95012  95017  95022  95027  95032 
95061   95066  95071   95075   95080 
95  109  95  114  95  119  95  124  95  129 
95158  95163  95168  95173  95177 

895 

896 
897 
898 
899 

95  182  95  187  95  192  95  197  95  202 
95231  95236  95240  95245  95250 
95279  95284  95289  95294  95299 
95328  95332  95337   95342  95347 
95376  95381  95386  95390  95395 

95207  95211   95216  95221   95226 
95255   95260  95265   95270  95274 
95303  95308  9S  313  95318  95323 
95352  95357  95361  95366  95371 
95400  95405   95410  95415   95419 

900 

95424  95429  95434  95439  95444 

95448  95453  95458  95463  95468 

N 

O  '       1          2          3         4 

56789 

850-900 


18 


900-950 


N 

01234 

56789 

9OO 

95424  95429  95434  95439  95444 

95448  95453  95458  95463  95468 

901 

95472  95477  95482  95487  95492 

95497  95501  95506  95511  95516 

902 

95521  95525  95530  95535   95540 

95545   95550  95554  95559  95564 

903 

95569  95574  95578  95583  95588 

95593   95598  95602  95607   95612 

904 

95617  95622  95626  95631  95636 

95641  95646  95650  95655   95660 

905 

95665   95670  95674  95679  95684 

95  689  95  694  95  698  95  703  95  708 

906 

95713  95718  95722  95727  95732 

95  737  95  742  95  746  95  751   95  756 

907 

95  761   95  766  95  770  95  775   95  780 

95  785   95  789  95  794  95  799  95  804 

908 

95809  95813  95818  95823  95828 

95832  95837  95842   95847   95852 

909 

95856  95861  95866  95871   95875 

95880  95885   95890  95895  95899 

910 

95904  95909  95914  95918  95923 

95  928  95  933  ^95  938  95  942  95  947 

911 

95952  95957  95961  95  966  95  971 

95976  95980  95985  95990  95995 

912 

95999  96004  96009  96014  96019 

96023   96028  96033   96038  96042 

913 

96047  96052  96057   96061   96066 

96071   96076  96080  96085   96090 

914 

96095   96099  96104  96109  96114 

96118  96123  96128  96133  96137 

915 

96142  96147   96152  96156  96161 

96166  96171   96175   96180  96185 

916 

96190  96194  96199  96204  96209 

96213   96218  96223   96227   96232 

917 

96237  96242  96246  96251   96256 

96261   96265   96270  96275   96280 

918 

96284  96289  96294  96298  96303 

96308  96313  96317  96322  96327 

919 

96332  96336  96341  96346  96350 

96355  96360  96365   96369  96374 

92O 

96379  96384  96388  96393  96398 

96402  96407  96412  96417  96421 

921 

96426  96431   96435   96440   96445 

96450  96454  96459  96464  96468 

922 

96473   96478  96483   96487   96492 

96497  96501   96506  96511   96515 

923 

96520  96525   96530  96534  96539 

96544  96548  96553   96558  96562 

924 

96567  96572  96577  96581   96586 

96591   96595  96600  96605   96609 

925 

96614  96619  96624  96628  96633 

96638  96642  96647  96652   96656 

926 

96661   96666  96670  96675   96680 

96685  .96689  96694   96699  96703 

927 

96708  96713  96717  96722  96727 

96731   96736  96741   96745  96750 

928 

96755   96759  96764  96769  96774 

96778  96783  96788  96792  96797 

929 

96802  96806  96811  96816  96820 

96825  96830  96834  96839  96844 

930 

96848  96853  96858  96862  96867 

96872  96876  96881  96886  96890 

931 

96895   96900  96904  96909  96914 

96918  96923   96928   96932  96937 

932 

96942  96946  96951   96956  96960 

96965    96970  96974  96979  96984 

933 

96988  96993   96997  97002  97007 

97011  97016  97021  97025   97030 

934 

97035   97039  97044  97049  97053 

97058  97063  97067  97072  97077 

935 

97081  97086  97090  97095   97100 

97104  97109  97114  97118  97123 

936 

97128  97132  97137  97142  97146 

97151  97155   97160  97165   97169 

937 

97174  97179  97183  97188  97192 

97197   97202  97206  97211   97216 

938 

97220  97225   97230  97234   97239 

97243   97248  97253   97257   97262 

939 

97267  97271   97276  97280  97285 

97290  97294  97299  97304  97308 

940 

97313  97317  97322  97327  97331 

97336  97340  97345   97350  97354 

941 

97359  97364  97368  97373   97377 

97382  97387  97391   97396  97400 

942 

97405  97410  97414  97419  97424 

97428  97433   97437   97442   97447 

943 

97451  97456  97460  97465   97470 

97474  97479  97483   97488  97493 

944 

97497  97502  97506  97511  97516 

97520  97525   97529  97534  97539 

945 

97543  97548  97552  97557  97562 

97566  97571   97575  97580  97585 

946 

97589  97594  97598  97603  97607 

97612  97617  97621    97626  97630 

947 

97635   97640  97644  97649  97653 

97658  97663   97667   97672  97676 

948 

97681   97685  97690  97695  97699 

97704  97708  97713   97717  97722 

949 

97727  97731   97736  97740  97745 

97749  97754  97759  97763   97768 

95O 

97772  97777  97782  97786  97791 

97795   97800  97804   97809  97813 

N 

O          1          2          3         4 

56789 

900-950 


950-1000 


19 


N 

O          1          2          3          4 

56789 

95O 

97772  97777  97782  97786  97791 

97795   97800  97804  97809  97813 

951 

97818  97823  97827  97832  97836 

97841  97845   97850  97855   97859 

952 

97864  97868  97873  97877  97882 

97886  97891  97896  97900  97903 

953 

97909  97914  97918  97923  97928 

97932  97937  97941   97946  97950 

954 

97955  97959  97964  97968  97973 

97978  97982  97987  97991  97996 

955 

98000  98005   98009  98014  98019 

98023  98028  98032  98037  98041 

956 

98046  98050  98055   98059  98064 

98068  98073  98078  98082  98087 

957 

98091  98096  98100  98105   98109 

98114  98118  98123  98127  98132 

958 

98137  98141  98146  98150  98155 

98159  98164  98168  98173  98177 

959 

98182  98186  98191  98195   98200 

98204   98209   98214  98218  98223 

96O 

98227  98232  98236  98241   98245 

982^0  98254  98259  98263  98268 

961 

98272  98277  98281  98286  98290 

98295   98299  98304  98308  98313 

962 

98318  98322  98327  98331  98336 

98340  98345   98349  98354  98358 

963 

98363  98367  98372  98376  98381 

98385   98390  98394  98399  98403 

964 

98408  98412  98417  98421  98426 

98430  98435   98439  98444  98448 

965 

98453  98457  98462  98466  98471 

98475  98480  98484  98489  98493 

966 

98498  98502  98507  98511  98516 

98520  98525   98529  98534  98538 

967 

98543  98547  98552  98556  98561 

98565   98570  98574  98579  98583 

968 

98588  98592  98597  98601   98605 

98610  98614  98619  98623  98628 

969 

98632  98637  98641  98646  98.650 

98655   98659  98664  986,68  98673 

970 

98677  98682  98686  98691   98695 

98700  98704  98709  98713  98717 

971 

98722  98726  98731  98735  98740 

98744  98749  98753  98758  98762 

972 

98767  98771  98776  98780  98784 

98789  98793  98798  98802  98807 

973 

98811  98816  98820  98825   98829 

98834  98838  98843  98847  98851 

974 

98856  98860  98865   98869  98874 

98878  98883  98887  98892  98896 

975 

98900  98905   98909  98914  98918 

98923  98927  98932  98936  98941 

976 

98945   98949  98954  98958  98963 

98967  98972  98976  98981   98985 

977 

98989  98994  98998  99003   99007 

99012  99016  99021   99025   99029 

978 

99034  99038  99043  99047  99052 

99056  99061   99065   99069  99074 

979 

99078  99083  99087  99092  99096 

99100  99105   99109  99114  99118 

980 

99123  99127  99131  99136  99140 

99145   99149  99154  99158  99162 

981 

99167  99171  99176  99180  99185 

99189  99193  99198  99202  99207 

982 

99211   99216  99220  99224  99229 

99233   99238  99242  99247   99251 

983 

99255   99260  99264  99269  99273 

99277  99282  99286  99291  99295 

984 

99300  99304  99308  99313   99317 

99  322  99  326  99  330  99  335  99  339 

985 

99344  99348  99352  99357  99361 

99366  99370  99374  99379  99383 

986 

99388  99392  99396  99401   99405 

99410  99414  99419  99423   99427 

987 

99432  99436  99441   99445   99449 

99454  99458  99463  99467  99471 

988 

99476  99480  99484  99489  99493 

99498  99502  99506  99511   99515 

989 

99520  99524  99528  99533   99537 

99542  99546  99550  99555   99559 

99O 

99564  99568  99572  99577  99581 

99585   99590  99594  99599  99603 

991 

99607  99612  99616  99621   99625 

99629  99634  99638  99642  99647 

992 

99651   99656  99660   99664  99669 

99673  99677  99682  99686  99691 

993 

99695   99699  99704  99708  99712 

99717  99721  99726  99730  99734 

994 

99739  99743  99747  99752  99756 

99760  99765   99769  99774  99778 

995 

99782  99787  99791  99795  99800 

99804  99808  99813  99817  99822 

996 

99826  99830  99835   99839  99843 

99848  99852  99856  99861   99865 

997 

99870  99874  99878  99883  99887 

99891  99896  99900  99904  99909 

998 

99913   99917   99922  99926  99930 

99935   99939  99944  99948  99952 

999 

99957  99961  99965  99970  99974 

99978  99983  99987  99991  99996 

1OOO 

00000  00004  00009  00013  00017 

00022  00026  00030  00  033  00039 

N 

O          1          2          3         4 

56789 

950-1000 


20     TABLE  II. -LOGARITHMS  OF  CONSTANTS. 


Circumference  of  th 
Circumference  of  th 
Circumference  of  th 
If  the  radius  r  =  1,  1 
TT  =  3.  14  159  265  3 

log 
2.55630250 
4.  33  445  375 
6.11260500 

0.  49  714  987 

e  Circle  in  minutes               ....  —       21  600 

e  Circle  in  seconds  —  1  296  000 

lalf  the  Circumferenc 
58  979  323  846  264  338 

e  of  the  Circle  is 
328  

Also: 
27r  =    6.28318531 
47r  =  12.56637061 
^=    1.57079633 

£=    1.04719755 
3 

i^=    4.18879020 
3 

£  =    0.78539816 
4 

£=    0.52359878 
i=    0.31830989 
~  =    0.15915494 
|=    0.95492966 
-=    1.27323954 

7T 

—  =    0.23873241 

4?r 

log 
0.  79  817  987 

1.  09  920  986 
0.  19  611  988 

0.  02  002  862 
0.  62  208  861 
9.  89  508  988  -  10 
9.  71  899  862  .-10 
9.  50  285  013  -  10 
9.  20  182  013  -  10 
9.  97  997  138  -  10 
0.10491012 
9.  37  791  139  -  10 

-r2  =  9.  86  960  440 
1-0.  10  132  US 
VTT  =  1.  77  245  385 

—  =  0.  56  418  958 
VT 

J^  =  0.  97  720  502 

J4  =1.12837917 
*/TT  =  1.46  459  189 

—  =  0.  68  278  406 
& 
^/7r2  =  2.  14  502  940 

|  /A  =  0.62  035  049 

\47T 

^  =  0.  80  599  598 

log 
0.99429975 

9.00570025-10 
0.24857494 
9.  75  142  506  -  10 

9.98998569-10 

0.  05  245  506 
0.16571662 
9.  83  428  338  -  10 

0.  33  143  325 
9.  79  263  713  -  10 

9.  90  633  287  -  10 

Arc  a,  whose  length  is  equal  to  the  radius  r,  is  : 
in  degrees    ...      a°      .     —  18°                 —5729577951°. 

log 
1.  75  812  263 

3.  53  627  388 
5.  31  442  513 

2.  05  915  263 
3.83730388 
5.  61  545  513 

8  24  187  737      10 

7T 

in  minutes             a'           —  10  ^°°            —  3  437  74  677' 

7T 

in  seconds             «"          -  648  °°°          -  ?H6  9.64  R06" 

Arc  2  a,  whose  lengt 
in  degrees  

in  minutes  
in  seconds  ..... 
If  the  radius  r  =  1, 

7T 

h  is  equal  to  twice  the  radius,  2  r,  is  : 
.  2  a°  .  .  .  .  =  —•  =  114.  59  155  903° 

2  a'             ^                      6  875  49  354f 

7T 

0  a"             1  2^6  °°^           41°  5°9  61°" 

7T 

ihe  length  of  the  arc  is  : 
1  .           -   ""                —  n  m  74";  -*9Q 

for  1  minute                                   ""     .                  0  00  0°9  089 

6.46372612      10 

.       a'                10  800 
for  1  second                                      T                     0  00  000  48  "5 

4  68557487      10 

"a"'"           648000" 
for  £  degree  ...-      *        -   —    ""                  —  000879.  66S 

7.  94  084  737  -  10 
616  969  61  2       10 

t      for  £  minute.  .  . 
for  $  second  .  .  . 
Sin  1"  in  the  unit  ci 

2a°             360 
"*                   Q  00014  544 

'2  a'              21600 
—          ""                 —  0  00  000  94^ 

4.38454487-10 
4.68557487-10 

"2a»  1296000  a00(X 
rcle  -  0.  00  000  485.  .  • 

21 


TABLE  III. 

THE  LOGAEITHMS 

OF  THE 

TRIGONOMETRIC   FUNCTIONS  : 

Prom  0°  to  0°  3',  or  89°  57'  to  90°,  for  every  second  ; 

From  0°  to  2°,  or  88°  to  90°,  for  every  ten  seconds  ; 

From  1°  to  89°,  for  every  minute, 

NOTE.    To  all  the  logarithms  —  10  is  to  be  appended. 

i               •                                     /\O                        log  tan  =  log  sin. 
lOg   SlU                                U                          log  cos  =  10.  00  000 

ft 

O'              1'             2' 

r  f 

r  t 

O'             1'             2' 

f  r 

o 

—         6.  46  373    6.  76  476 

6O 

30 

6.16270    6.63982    6.86167 

30 

1 

4.  68  557    6.  47  090    6.  76  836 

59 

31 

6.17694    6.64462    6.86455 

29 

2 

4.  98  660    6.  47  797    6.  77  193 

58 

32 

6.19072    6.64936    6.86742 

28 

3 

5.16270    6.48492    6.77548 

57 

33 

6.20409    6.65406    6.87027 

27 

4 

5.  28  763    6.  49  175    6.  77  900 

56 

34 

6.21705    6.65870    6.87310 

26 

5 

5.  38  454    6.  49  849    6.  78  248 

55 

35 

6.22964    6.66330    6.87591 

25 

6 

5.46373    6.50512    6.78595 

54 

36 

6.24188    6.66785    6.87870 

24 

7 

5.53067    6.51165     6.78938 

53 

37 

6.25378    6.67235    6.88147 

23 

8 

5.58866    6.51808    6.79278 

52 

38 

6.26536    6.67680    6.88423 

22 

9 

5.  63  982    6.  52  442    6.  79  616 

51 

39 

6.27664    6.68121    6.88697 

21 

10 

5.68557    6.53067    6.79952 

5O 

40 

6.28763    6.68557    6.88969 

2O 

11 

5.72697    6.53683     6.80285 

49 

41 

6.29836    6.68990    6.89240 

19 

12 

5.76476    6.54291    6.80615 

48 

42 

6.30882    6.69418    6.89509 

18 

13 

5.  79  952    6.  54  890    6.  80  943 

47 

43 

6.31904    6.69841    6.89776 

17 

14 

5.  83  170    6.  55  481    6.  81  268 

46 

44 

6.32903    6.70261    6.90042 

16 

15 

5.  86  167    6.  56  064    6.  81  591 

45 

45 

6.33879    6.70676    6.90306 

15 

16 

5.88969    6.56639    6.81911 

44 

46 

6.34833    6.71088    6.90568 

14 

17 

5.  91  602    6.  57  207    6.  82  230 

43 

47 

6.35767    6.71496    6.90829 

13 

18 

5.94085     6.57767    6.82545 

42 

48 

6.36682    6.71900    6.91088 

12 

19 

5.96433    6.58320    6.82859 

41 

49 

6.37577    6.72300    6.91346 

11 

20 

5.98660    6.58866    6.83170 

4O 

5O 

6.38454    6.72697    6.91602 

10 

21 

6.00779    6.59406    6.83479 

39 

51 

6.39315    6.73090    6.91857 

9 

22 

6.02800    6.59939    6.83786 

'  38 

52 

6.40158    6.73479    6.92110 

8 

23 

6.  04  730    6.  60  465     6.  84  091 

37 

53 

6.40985    6.73865    6.92362 

7 

24 

6.  06  579    6.  60  985     6.  84  394 

36 

54 

6.41797    6.74248    6.92612 

6 

25 

6.08351    6.61499    6.84694 

35 

55 

6.42594    6.74627    6.92861 

5 

26 

6.  10  055     6.  62  007    6.  84  993 

34 

56 

6.43376    6.75003    6.93109 

4 

27 

6.  11  694    6.  62  509    6.  85  289 

33 

57 

6.44145    6.75376    6.93355 

3 

28 

6.13273    6.63006    6.85584 

32 

58 

6.44900    6.75746    6.93599 

2 

29 

6.14797    6.63496    6.85876 

31 

59 

6.45643    6.76112    6.93843 

1 

30 

6.16270    6.63982    6.86167 

30 

60 

6.46373    6.76476    6.94085. 

0 

ff 

59'           58'          57' 

tf 

rr 

59'         58'          57' 

tt 

log  COt  =  log  008 

log  sin  - 10,  00  000 


89° 


log  cos 


22 


0° 


t  tt 

log  sin      log  tan      log  cos 

//    t 

t  tt 

log  sin      log  tan      log  cos 

tt  r 

O    0 

—              —         10.00000 

06O 

1OO 

7.  46  373     7.  46  373     10.00000 

05O 

10 

5.68557    5.68557     10.00000 

50 

10 

7.  47  090    7.  47  091     10.00000 

50 

20 

5.98660    5.98660     10.00000 

40 

20 

7.  47  797    7.  47  797     10.00000 

40 

30 

6.  16  270    6.  16  270     10.00000 

30 

30 

7.  48  491     7.  48  492     10.00000 

30 

40 

6.  28  763    6/28  763     10.00000 

20 

40 

7.  49  175     7.  49  176    10.00000 

20 

50 

6.38454    6.38454     10.00000 

10 

50 

7.  49  849    7.  49  849     10.00000 

10 

1     0 

6.46373    6.46373     10.00000 

059 

110 

7.50512     7.50512     10.00000 

049 

10 

6.  53  067     6.  53  067     10.00000 

50 

10 

7.51165     7.51165     10.00000 

50 

20 

6.58866    6.58866     10.00000 

40 

20 

7.  51  808    7.  51  809    10.00000 

40 

30 

6.63982    6.63982     10.00000 

30 

30 

7.  52  442     7.  52  443     10.00000 

30 

40 

6.  68  557    6.  68  557     10.00000 

20 

40 

7.53067    7.53067     10.00000 

20 

50 

6.  72  697    6.  72  697     10.00000 

10 

50 

7.  53  683     7.  53  683     10.00000 

10 

2    0 

6.76476    6.76476    10.00000 

058 

120 

7.  54  291     7.  54  291     10.00000 

048 

10 

6.  79  952    6.  79  952     10.00000 

50 

10 

7.  54  890    7.  54  890    10.00000 

50 

20 

6.83170    6.83170    10.00000 

40 

20 

7.  55  481     7.  55  481     10.00000 

40 

30 

6.  86  167    6.  86  167     10.00000 

30 

30 

7.  56  064    7.  56  064    10.00000 

30 

40 

6.  88  969    6.  88  969     10.00000 

20 

40 

7.  56  639    7.  56  639    10.00000 

20 

50 

6.91602    6.91602     10.00000 

10 

50 

7.  57  206    7.  57  207     10.00000 

10 

3    0 

6.  94  085     6.  94  085     10.00000 

057 

130 

7.  57  767    7.  57  767     10.00000 

047 

10 

6.96433     6.96433     10.00000 

50 

10 

7.58320    7.58320    10.00000 

50 

20 

6.98660    6.98661     10.00000 

40 

20 

7.  58  866    7.  58  867     10.00000 

40 

30 

7.  00  779    7.  00  779    10.00000 

30 

30 

7.  59  406     7.  59  406     10.00000 

30 

40 

7.02800    7.02800     10.00000 

20 

40 

7.  59  939     7.  59  939     10.00000 

20 

50 

7.  04  730    7.  04  730     10.00000 

10 

50 

7.  60  465     7.  60  466    10.00000 

10 

4    0 

7.  06  5  79    7.  06  5  79     10.00000 

056 

140 

7.60985     7.60986     10.00000 

046 

10 

7.08351     7.08352     10.00000 

50 

10 

7.  61  499     7.  61  500     10.00000 

50 

20 

7.10055     7.10055     10.00000 

40 

20 

7.  62  007    7.  62  008     10.00000 

40 

30 

7.  11  694    7.  11  694     10.00000 

30 

30 

7.62509     7.62510     10.00000 

30 

40 

7.13273     7.13273     10.00000 

20 

40 

7.  63  006     7.  63  006     10.00000 

20 

50 

7.  14  797    7.  14  797     10.00000 

10 

50 

7.  63  496     7.  63  497     10.00000 

10 

5    0 

7.16270    7.16270     10.00000 

055 

150 

7.63982     7.63982     10.00000 

045 

10 

7.  17  694     7.  17  694     10.00000 

50 

10 

7.  64  461     7.  64  462     10.00000 

50 

20 

7.  19  072     7.  19  073     10.00000 

40 

20 

7.  64  936    7.  64  937     10.00000 

40 

30 

7.20409     7.20409     10.00000 

30 

30 

7.65406     7.65406     10.00000 

30 

40 

7.  21  705     7.  21  705     10.00000 

20 

40 

7.65870    7.65871     10.00000 

20 

50 

7.  22  964    7.  22  964     10.00000 

10 

50 

7.  66  330    7.  66  330     10.00000 

10 

6    0 

7.  24  188    7.  24  188     10.00000 

054 

160 

7.  66  784    7.  66  785     10.00000 

044 

10 

7.  25  378    7.  25  378     10.00000 

50 

10 

7.  67  235     7.  67  235     10.00000 

50 

20 

7.26536    7.26536     10.00000 

40 

20 

7.67680    7.67680    10.00000 

40 

30 

7.27664     7.27664     10.00000 

30 

30 

7.  68  121     7.  68  121     10.00000 

30 

40 

7.  28  763     7.  28  764    10.00000 

20 

40 

7.  68  557     7.  68  558      9.99999 

20 

50 

7.  29  836    7.  29  836     10.00000 

10 

50 

7.68989    7.68990      9.99999 

10 

7    0 

7.  30  882    7.  30  882    10.00000 

053 

170 

7.69417     7.69418    9.99999 

043 

10 

7.  31  904    7.  31  904     10.00000 

50 

10 

7.69841     7.69842    9.99999 

50 

20 

7.32903     7.32903     10.00000 

40 

20 

7.  70  261     7.  70  261    9.  99  999 

40 

30 

7.  33  879    7.  33  879     10.00000 

30 

30 

7.  70  676    7.  70  677    9.  99  999 

30 

40 

7.  34  833     7.  34  833     10.00000 

20 

40 

7.71088    7.71088    9.99999 

20 

50 

7.35767    7.35767     10.00000 

10 

50 

7.71496    7.71496    9.99999 

10 

8    0 

7.  36  682    7.  36  682    10.00000 

052 

180 

7.71900    7.71900    9.99999 

042 

10 

7.37577    7.37577     10.00000 

50 

10 

7.  72  300    7.  72  301    9.  99  999 

50 

20 

7.  38  454    7.  38  455     10.00000 

40 

20 

7.  72  697     7.  72  697    9.  99  999 

40 

30 

7.39314    7.39315     10.00000 

30 

30 

7.73090     7.73090    9.99999 

30 

40 

7.40158    7.40158     10.00000 

20 

40 

7.73479    7.73480    9.99999 

20 

50 

7.  40  985     7.  40  985     10.00000 

10 

50 

7.73865     7.73866    9.99999 

10 

9    0 

7.  41  797    7.  41  797     10.00000 

051 

190 

7.74248    7.74248    9.99999 

041 

10 

7.  42  594    7.  42  594     10.00000 

50 

10 

7.74627     7.74628    9.99999 

50 

20 

7.43376    7.43376    10.00000 

40 

20 

7.75003    7.75004    9.99999 

40 

30 

7.  44  145     7.  44  145     10.00000 

30 

30 

7.75376    7.75377    9.99999 

30 

40 

7.  44  900    7.  44  900     10.00000 

20 

40 

7.  75  745     7.  75  746    9.  99  999 

20 

50 

7.45643     7.45643     10.00000 

10 

50 

7.76112    7.76113    9.99999 

10 

1OO 

7.46373    7.46373     10.00000 

05O 

2O  0 

7.76475     7.76476    9.99999 

04O 

f  tt 

log  cos      log  cot      log  sin 

ft   t 

t  ft 

log  cos      log  cot      log  sin 

ft  f 

89C 


23 


t  " 

log  sin      log  tan      log  cos 

/;    t 

r    ft 

log  sin      log  tan      log  cos 

rt  t 

20  o 

7.76475     7.76476    9.99999 

040 

3OO 

7.94084    7.94086    9.99998 

030 

10 

7.  76  836    7.  76  837    9.  99  999 

50 

10 

7.94325     7.94326    9.99998 

50 

20 

7.77193    7.77194    9.99999 

40 

20 

7.94564     7.94566    9.99998 

40 

30 

7.77548    7.77549    9.99999 

30 

30 

7.94802     7.94804    9.99998 

30 

40 

7.77899    7.77900    9.99999 

20 

40 

7.95039     7.95040    9.99998 

20 

50 

7.78248    7.78249    9.99999 

10 

50 

7.95274    7.95276    9.99998 

10 

210 

7.78594    7.78595    9.99999 

039 

310 

7.95508    7.95510    9.99998 

029 

10 

7.78938    7.78938    9.99999 

50 

10 

7.95741     7.95743    9.99998 

50 

20 

7.79278     7.79279    9.99999 

40 

20 

7.95973     7.95974    9.99998 

40 

30 

7.79616    7.79617    9.99999 

30 

30 

7.  96  203     7.  96  205     9.  99  998 

30 

40 

7.79952    7.79952    9.99999 

20 

40 

7.96432     7.96434    9.99998 

20 

50 

7.  80  284    7.  80  285    9.  99  999 

10 

50 

7.  96  660    7.  96  662    9.  99  998 

10 

220 

7.80615     7.80615    9.99999 

038 

320 

7.96887    7.96889    9.99998 

028 

10 

7.  80  942     7.  80  943    9.  99  999 

50 

10 

7.97113     7.97114    9.99998 

50 

20 

7.81268    7.81269    9.99999 

40 

20 

7.97337    7.97339    9.99998 

40 

30 

7.  81  591     7.  81  591    9.  99  999 

30 

30 

7.97560    7.97562    9.99998 

30 

40 

7.81911     7.81912    9.99999 

20 

40 

7.97782    7.97784    9.99998 

20 

50 

7.82229    7.82230    9.99999 

10 

50 

7.  98  003     7.  98  005    9.  99  998 

10 

230 

7.82545     7.82546    9.99999 

037 

330 

7.  98  223    7.  98  225    9.  99  998 

027 

10 

7.82859     7.82860    9.99999 

50 

10 

7.98442     7.98444    9.99998 

50 

20 

7.83170    7.83171    9.99999 

40 

20 

7.  98  660     7.  98  662    9.  99  998 

40 

30 

7.83479    7.83480    9.99999 

30 

30 

7.98876    7.98878    9.99998 

30 

40 

7.83786    7.83787    9.99999 

20 

40 

7.99092     7.99094    9.99998 

20 

50 

7.  84  091     7.  84  092    9.  99  999 

10 

50 

7.  99  306    7.  99  308    9.  99  998 

10 

240 

7.84393    7.84394    9.99999 

036 

340 

7.  99  520    7.  99  522    9.  99  998 

026 

10 

7.  84  694    7.  84  695     9.  99  999 

50 

10 

7.99732     7.99734    9.99998 

50 

20 

7.  84  992     7.  84  994    9.  99  999  j  40 

20 

7.99943     7.99946    9.99998 

40 

30 

7.  85  289    7.  85  290    9.  99  999  !  30 

30 

8.00154    8.00156    9.99998 

30 

40 

7.85583     7.85584    9.99999 

20 

40 

8.  00  363     8.  00  365     9.  99  998 

20 

50 

7.  85  876    7.  85  877    9.  99  999 

10 

50 

8.00571     8.00574    9.99998 

10 

250 

7.  86  166    7.  86  167    9.  99  999 

035 

350 

8.00779    8.00781    9.99998 

025 

10 

7.86455     7.86456    9.99999 

50 

10 

8.  00  985     8.  00  987    9.  99  998 

50 

20 

7.  86  741     7.  86  743    9.  99  999 

40 

20 

8.  01  190    8.  01  193    9.  99  998 

40 

30 

7.87026    7.87027    9.99999 

30 

30 

8.  01  395     8.  01  397    9.  99  998 

30 

40 

7.87309    7.87310    9.99999 

20 

40 

8.01598    8.01600    9.99998 

20 

50 

7.  87  590    7.  87  591    9.  99  999 

10 

50 

8.01801     8.01803    9.99998 

10 

260 

7.  87  870    7.  87  871    9.  99  999 

034 

360 

8.  02  002    8.  02  004    9.  99  998 

024 

10 

7.88147     7.88148    9.99999 

50 

10 

8.02203     8.02205     9.99998 

50 

20 

7.88423     7.88424    9.99999 

40 

20 

8.  02  402    8.  02  405     9.  99  998 

40 

30 

7.  88  697    7.  88  698    9.  99  999 

30 

30 

8.  02  601     8.  02  604    9.  99  998 

30 

40 

7.88969    7.88970    9.99999 

20 

40 

8.  02  799    8.  02  801     9.  99  998 

20 

50 

7.  89  240    7.  89  241    9.  99  999 

10 

50 

8.02996    8.02998    9.99998 

10 

270 

7.89509    7.89510    9.99999 

033 

370 

8.03192    8.03194    9.99997 

023 

10 

7.89776    7.89777    9.99999 

50 

10 

8.03387     8.03390    9.99997 

50 

20 

7.  90  041     7.  90  043    9.  99  999 

40 

20 

8.03581    8.03584    9.99997 

40 

30 

7.  90  305     7.  90  307    9.  99  999 

30 

30 

8.  03  775     8.  03  777    9.  99  997 

30 

40 

7.  90  568     7.  90  569    9.  99  999 

20 

40 

8.03967    8.03970    9.99997 

20 

50 

7.90829    7.90830    9.99999 

10 

50 

8.  04  159    8.  04  162    9.  99  997 

10 

280 

7.91088    7.91089    9.99999 

032 

380 

8.04350    8.04353    9.99997 

022 

10 

7.  91  346    7.  91  347    9.  99  999 

50 

10 

8.  04  540    8.  04  543     9.  99  997 

50 

20 

7.  91  602     7.  91  603     9.  99  999 

40 

20 

8.04729    8.04732    9.99997 

40 

30 

7.91857    7.91858    9.99999 

30 

30 

8.04918    8.04921     9.99997 

30 

40 

7.92110     7.92111     9.99998 

20 

40 

8.  05  105     8.  05  108    9.  99  997 

20 

50 

7.92362    7.92363    9.99998 

10 

50 

8.  05  292    8.  05  295     9.  99  997 

10 

290 

7.  92  612    7.  92  613    9.  99  998 

031 

390 

8.  05  478    8.  05  481    9.  99  997 

021 

10 

7.92861     7.92862    9.99998 

50 

10 

8.05663    8.05666    9.99997 

50 

20 

7.93108     7.93110    9.99998 

40 

20 

8.05848    8.05851    9.99997 

40 

30 

7.93354    7.93356    9.99998 

30 

30 

8.06031     8.06034    9.99997 

30 

40 

7.  93  599    7.  93  601     9.  99  998 

20 

40 

8.06214    8.06217    9.99997 

20 

50 

7.93842    7.93844    9.99998 

10 

50 

8.06396    8.06399    9.99997 

10 

3OO 

7.94084    7.94.086    9.99998 

030 

4OO 

8.06578    8.06581    9.99997 

020 

/  tt 

log  cos      log  cot      log  sin 

t?   f 

t  ff 

log  cos      log  cot      log  sin 

tr  r 

24 


f  ft 

log  sin     log  tan      log  cos 

ff    t 

t  tf 

log  sin      log  tan      log  cos 

ff  f 

400 

8.06578    8.06581     9.99997 

02O 

5OO 

8.16268    8.16273    9.99995 

01O 

10 

8.  06  758    8.  06  761    9.  99  997 

50 

10 

8.16413    8.16417    9.99995 

50 

20 

8.06938    8.06941     9.99997 

40 

20 

8.16557    8.16561     9.99995 

40 

30 

8.07117    8.07120    9.99997 

30 

30 

8.  16  700    8.  16  705    9.  99  995 

30 

40 

8.07295     8.07299    9.99997 

20 

40 

8.16843     8.16848    9.99995 

20 

50 

8.07473    8.07476    9.99997 

10 

50 

8.16986    8.16991    9.99995 

10 

410 

8.07650    8.07653    9.99997 

019 

510 

8.17128    8.17133    9.99995 

0    9 

10 

8.07826    8.07829    9.99997 

50 

10 

8.  17  270    8.  17  275     9.  99  995 

50 

20 

8.08002    8.08005     9.99997 

40 

20 

8.17411     8.17416    9.99995 

40 

30 

8.08176    8.08180    9.99997 

30 

30 

8.17552    8.17557    9.99995 

30 

40 

8.08350    8.08354    9.99997 

20 

40 

8.  17  692    8.  17  697    9.  99  995 

20 

50 

8.08524    8.08527    9.99997 

10 

50 

8.17832    8.17837    9.99995 

10 

420 

8.08696    8.08700    9.99997 

018 

520 

8.17971    8.17976    9.99995 

0     8 

10 

8.08868    8.08872    9.99997 

50 

10 

8.18110    8.18115    9.99995 

50 

20 

8.09040    8.09043    9.99997 

40 

20 

8.18249    8.18254    9.99995 

40 

30 

8.09210    8.09214    9.99997 

30 

30 

8.  18  387    8.  18  392    9.  99  995 

30 

40 

8.09380    8.09384    9.99997 

20 

40 

8.  18  524    8.  18  530    9.  99  995 

20 

50 

8.09550    8.09553    9.99997 

10 

50 

8.  18  662    8.  18  667    9.  99  995 

10 

430 

8.09718    8.09722    9.99997 

017 

530 

8.  18  798    8.  18  804    9.  99  995 

0     7 

10 

8.09886    8.09890    9.99997 

50 

10 

8.18935     8.18940    9.99995 

50 

20 

8.10054    8.10057    9.99997 

40 

20 

8.  19  071    8.  19  076    9.  99  995 

40 

30 

8.10220    8.10224    9.99997 

30 

30 

8.  19  206    8.  19  212    9.  99  995 

30 

40 

8.10386    8.10390    9.99997 

20 

40 

8.19341     8.19347    9.99995 

20 

50 

8.  10  552    8.  10  555     9.  99  996 

10 

50 

8.19476    8.19481    9.99995 

10 

440 

8.10717    8.10720    9.99996 

016 

540 

8.  19  610    8.  19  616    9.  99  995 

0    6 

10 

8.  10  881     8.  10  884    9.  99  996 

50 

10 

8.  19  744    8.  19  749    9.  99  995 

50 

20 

8.11044    8.11048    9.99996 

40 

20 

8.  19  877    8.  19  883    9.  99  995 

40 

30 

8.11207    8.11211     9.99996 

30 

30 

8.20010    8.20016    9.99995 

30 

40 

8.11370    8.11373    9.99996 

20 

40 

8.20143     8.20149    9.99995 

20 

50 

8.11531     8.11535     9.99996 

10 

50 

8.  20  275     8.  20  281     9.  99  994 

10 

450 

8.11693    8.11696    9.99996 

015 

550 

8.20407    8.20413    9.99994 

0    5 

10 

8.11853     8.11857    9.99996 

50 

10 

8.20538    8.20544    9.99994 

50 

20 

8.12013     8.12017    9.99996 

40 

20 

8.  20  669    8.  20  675     9.  99  994 

40 

30 

8.12172    8.12176    9.99996 

30 

30 

8.20800    8.20806    9.99994 

30 

40 

8.  12  331     8.  12  335     9.  99  996 

20 

40 

8.20930    8.20936    9.99994 

20 

50 

8.  12  489    8.  12  493    9.  99  996 

10 

50 

8.21060    8.21066    9.99994 

10 

460 

8.12647    8.12651     9.99996 

014 

560 

8.  21  189    8.  21  195    9.  99  994 

0    4 

10 

8.12804    8.12808    9.99996 

50 

10 

8.21319    8.21324    9.99994 

50 

20 

8.  12  961     8.  12  965     9.  99  996 

40 

20 

8.21447    8.21453    9.99994 

40 

30 

8.13117    8.13121    9.99996 

30 

30 

8.21576    8.21581     9.99994 

30 

40 

8.13272    8.13276    9.99996 

20 

40 

8.21703     8.21709    9.99994 

20 

50 

8.13427    8.13431     9.99996 

10 

50 

8.  21  831     8.  21  837    9.  99  994 

10 

470 

8.13581     8.13585     9.99996 

013 

570 

8.21958    8.21964    9.99994 

0    3 

10 

8.  13  735    8.  13  739    9.  99  996 

50 

10 

8.22085     8.22091     9.99994 

50 

20 

8.  13  888    8.  13  892    9.  99  996 

40 

20 

8.22211     8.22217    9.99994 

40 

30 

8.  14  041     8.  14  045     9.  99  996 

30 

30 

8.  22  337    8.  22  343    9.  99  994 

30 

40 

8.  14  193    8.  14  197    9.  99  996 

20 

40 

8.  22  463     8.  22  469    9.  99  994 

20 

50 

8.  14  344    8.  14  348    9.  99  996 

10 

50 

8.22588    8.22595     9.99994 

10 

480 

8.14495    8.14500    9.99996 

012 

580 

8.22713    8.22720    9.99994 

0    2 

10 

8.14646    8.14650    9.99996 

50 

10 

8.22838    8.22844    9.99994 

50 

20 

8.14796    8.14800    9.99996 

40 

20 

8.22962    8.22968    9.99994 

40 

30 

8.14945     8.14950    9.99996 

30 

30 

8.23086    8.23092    9.99994 

30 

40 

8.15094    8.15099    9.99996 

20 

40 

8.23210    8.23216    9.99994 

20 

50 

8.15243    8.15247    9.99996 

10 

50 

8.23333    8.23339    9.99994 

10 

490 

8.15391    8.15395    9.99996 

Oil 

590 

8.23456    8.23462    9.99994 

0     1 

10 

8.15538    8.15543    9.99996 

50 

10 

8.23578    8.23585     9.99994 

50 

20 

8.15685    8.15690    9.99996 

40 

20 

8.23700    8.23707    9.99994 

40 

30 

8.15832    8.15836    9.99996 

30 

30 

8.23822    8.23829    9.99993 

30 

40 

8.15978    8.15982    9.99995 

20 

40 

8.23944    8.23950    9.99993 

20 

50 

8.  16  123     8.  16  128    9.  99  995 

10 

50 

8.  24  065    8.  24  071    9.  99  993 

10 

5OO 

8.  16  268    8.  16  273    9.  99  995 

01O 

6OO 

8.24186    8.24192    9.99993 

0     O 

/  tt 

log  cos      log  cot      log  sin 

ft   r 

f  n 

log  cos      log  cot      log  sin 

ff  f 

89< 


25 


f  ft 

log  sin      log  tan      log  cos 

//    t 

iff 

log  sin      log  tan      log  cos 

ff  f 

O    0 

8.24186    8.24192    9.99993 

06O 

1OO 

8.  30  879    8.  30  888    9.  99  991 

05O 

10 

8.24306    8.24313    9.99993 

50 

10 

8.  30  983     8.  30  992    9.  99  991 

50 

20 

8.24426    8.24433    9.99993 

40 

20 

8.31086    8.31095     9.99991 

40 

30 

8.24546    8.24553    9.99993 

30 

30 

8.31188    8.31198    9.99991 

30 

40 

8.  24  665     8.  24  672    9.  99  993 

20 

40 

8.  31  291     8.  31  300    9.  99  991 

20 

50 

8.24785     8.24791     9.99993 

10 

50 

8.31393     8.31403     9.99991 

10 

1     0 

8.24903     8.24910    9.99993 

059 

110 

8.31495     8.31505     9.99991 

049 

10 

8.  25  022    8.  25  029    9.  99  993 

50 

10 

8.31597     8.31606    9.99991 

50 

20 

8.  25  140    8.  25  147    9.  99  993 

40 

20 

8.31699     8.31708    9.99991 

40 

30 

8.  25  258    8.  25  265     9.  99  993 

30 

30 

8.31800    8.31809    9.99991 

30 

40 

8.  25  375     8.  25  382    9.  99  993 

20 

40 

8.31901     8.31911     9.99991 

20 

50 

8.  25  493     8.  25  500    9.  99  993 

10 

50 

8.  32  002     8.  32  012    9.  99  991 

10    * 

2    0 

8.25609    8.25616    9.99993 

058 

120 

8.32103     8.32112    9.99990 

048 

10 

8.25726    8.25733     9.99993 

50 

10 

8.  32  203     8.  32  213     9.  99  990 

50 

20 

8.25842    8.25849    9.99993 

40 

20 

8  32  303     8.  32  313     9.  99  990 

40 

30 

8.25958    8.25965     9.99993 

30 

30 

8.  32  403     8.  32  413    9.  99  990 

30 

40 

8.  26  074    8.  26  081     9.  99  993 

20 

40 

8.32503     8.32513    9.99990 

20 

50 

8.  26  189    8.  26  196    9.  99  993 

10 

50 

8.  32  602     8.  32  612    9.  99  990 

10 

3    0 

8.26304    8.26312    9.99993 

057 

130 

8.32702     8.32711     9.99990 

047 

10 

8.26419    8.26426    9.99993 

50 

10 

8.  32  801     8.  32  811     9.  99  990 

50 

20 

8.  26  533     8.  26  541     9.  99  993 

40 

20 

8.  32  899    8.  32  909    9.  99  990 

40 

30 

8.  26  648    8.  26  655     9.  99  993 

30 

30 

8.  32  998    8.  33  008    9.  99  990 

30 

40 

8.26761     8.26769    9.99993 

20 

40 

8.33096    8.33106    9.99990 

20 

50 

8.26875     8.26882    9.99993 

10 

50 

8.  33  195     8.  33  205     9.  99  990 

10 

4    0 

8.  26  988    8.  26  996    9.  99  992 

056 

140 

8.  33  292     8.  33  302    9.  99  990 

046 

10 

8.  27  101     8.  27  109    9.  99  992 

50 

10 

8.33390    8.33400    9.99990 

50 

20 

8.  27  214    8.  27  221     9.  99  992 

40 

20 

8.  33  488    8.  33  498    9.  99  990 

40 

30 

8.  27  326    8.  27  334    9.  99  992 

30 

30 

8.33585     8.33595     9.99990 

30 

40 

8.  27  438    8.  27  446    9.  99  992 

20 

40* 

8.  33  682     8.  33  692    9.  99  990 

20 

50 

8.27550    8.27558    9.99992 

10 

50 

8.  33  779    8.  33  789    9.  99  990 

10 

5    0 

8.27661     8.27669    9.99992 

055 

150 

8.33875     8.33886    9.99990 

045 

10 

8.  27  773     8.  27  780    9.  99  992 

50 

10 

8.  33  972     8.  33  982    9.  99  990 

50 

20 

8.  27  883     8.  27  891     9.  99  992 

40 

20 

8.  34  068    8.  34  078    9.  99  990 

40 

30 

8.  27  994    8.  28  002    9.  99  992 

30 

30 

8.  34  164     8.  34  174    9.  99  990 

30 

40 

8.28104    8.28112    9.99992 

20 

40 

8.34260     8.34270    9.99989 

20 

50 

8.  28  215     8.  28  223     9.  99  992 

10 

50 

8.34355     8.34366    9.99989 

10 

6    0 

8.  28  324    8.  28  332    9.  99  992 

054 

160 

8.  34  450    8.  34  461     9.  99  989 

044 

10 

8.  28  434    8.  28  442    9.  99  992 

50 

10 

8.34546    8.34556    9.99989 

50 

20 

8.28543     8.28551     9.99992 

40 

20 

8.34640    8.34651     9.99989 

40 

30 

8.  28  652     8.  28  660    9.  99  992 

30 

30 

8.34735     8.34746    9.99989 

30 

40 

8.  28  761     8.  28  769    9.  99  992 

20 

40 

8.34830    8.34840    9.99989 

20 

50 

8.  28  869    8.  28  877     9.  99  992 

10 

50 

8.  34  924     8.  34  935     9.  99  989 

10 

7    0 

8.  28  977     8.  28  986    9.  99  992 

053 

170 

8.35018     8.35029    9.99989 

043 

10 

8.  29  085     8.  29  094     9.  99  992 

50 

10 

8.35112     8.35123     9.99989 

50 

20 

8.  29  193     8.  29  201     9.  99  992 

40 

20 

8.  35  206     8.  35  217    9.  99  989 

40 

30 

8.  29  300    8.  29  309    9.  99  992 

30 

30 

8.35299     8.35310    9.99989 

30 

40 

8.29407     8.29416    9.99992 

20 

40 

8.  35  392     8.  35  403     9.  99  989 

20 

50 

8.29514    8.29523     9.99992 

10 

50 

8.  35  485     8-.  35  497    9.  99  989 

10 

8    0 

8.  29  621     8.  29  629    9.  99  992 

052 

180 

8.35578    8.35590    9.99989 

042 

10 

8.29727    8.29736    9.99991 

50 

10 

8.  35  671     8.  35  682    9.  99  989 

50 

20 

8.  29  833     8.  29  842    9.  99  991 

40 

20 

8.  35  764    8.  35  775    9.  99  989 

40 

30 

8.  29  939    8.  29  947    9.  99  991 

30 

30 

8.  35  856    8.  35  867    9.  99  989 

30 

40 

8.  30  044    8.  30  053     9.  99  991 

20 

40 

8.35948     8.35959    9.99989 

20 

50 

8.30150    8.30158    9.99991 

10 

50 

8.36040    8.36051     9.99989 

10 

9    0 

8.  30  255     8.  30  263     9.  99  991 

051 

190 

8.36131     8.36143    9.99989 

041 

10 

8.  30  359    8.  30  368    9.  99  991 

50 

10 

8.36223     8.36235     9.99988 

50 

20 

8.  30  464    8.  30  473     9.  99  991 

40 

20 

8.36314     8.36326    9.99988 

40 

30 

8.  30  568    8.  30  577     9.  99  991 

30 

30 

8.  36  405     8.  36  417    9.  99  988 

30 

40 

8.  30  672    8.  30  681     9.  99  991 

20 

40 

8.36496     8.36508    9.99988 

20 

50 

8.  30  776    8.  30  785     9.  99  991 

10 

50 

8.  36  587    8.  36  599    9.  99  988 

10 

1OO 

8.30879    8.30888    9.99991 

05O 

2O  0 

8.36678    8.36689    9.99988 

040 

f  tr 

log  cos      log  cot      log  sin 

tf   f 

f  t  f 

log  cos      log  cot      log  sin 

rf  f 

88C 


26 


f  ft 

log  sin      log  tan      log  cos 

n   t 

t  tr 

log  sin      log  tan     log  cos 

ff  r 

20  o 

8.36678    8.36689    9.99988 

04O 

3OO 

8.41792    8.41807    9.99985 

03O 

10 

8.36768    8.36780    9.99988 

50 

10 

8.  41  872    8.  41  887    9.  99  985 

50 

20 

8.36858    8.36870    9.99988 

40 

20 

8.  41  952    8.  41  967     9.  99  985 

40 

30 

8.36948    8.36960    9.99988 

30 

30 

8.  42  032    8.  42  048    9.  99  985 

30 

40 

8.37038    8.37050    9.99988 

20 

40 

8.42112    8.42127     9.99985 

20 

50 

8.37128    8.37140    9.99988 

10 

50 

8.  42  192    8.  42  207    9.  99  985 

10 

210 

8.37217    8.37229    9.99988 

039 

310 

8.  42  272    8.  42  287    9.  99  985 

029 

10 

8,37306    8.37318    9.99988 

50 

10 

8.42351     8.42366    9.99985 

50 

20 

8.37395     8.37408    9.99988 

40 

20 

8.42430    8.42446    9.99985 

40 

30 

8.37484    8.37497    9.99988 

30 

30 

8.42510     8.42525     9.99985 

30 

40 

8.37573    8.37585    9.99988 

20 

40 

8.  42  589     8.  42  604    9.  99  985 

20 

50 

8.37662    8.37674    9.99988 

10 

50 

8.  42  667    8.  42  683    9.  99  985 

10 

220 

8.37750    8.37762    9.99988 

038 

320 

8.42746    8.42762    9.99984 

028 

10 

8.37838    8.37850    9k  99  988 

50 

10 

8.42825     8.42840    9.99984 

50 

20 

8.37926    8.37938    9.99988 

40 

20 

8.  42  903     8.  42  919    9.  99  984 

40 

30 

8.38014    8.38026    9.99987 

30 

30 

8.  42  982     8.  42  997    9.  99  984 

30 

40 

8.38101    8.38114    9.99987 

20 

40 

8.43060    8.43075     9.99984 

20 

50 

8.38189    8.38202    9.99987 

10 

50 

8.43138    8.43154    9.99984 

10 

230 

8.38276    8.38289    9.99987 

037 

330 

8.  43  216    8.  43  232    9.  99  984 

027 

10 

8.38363    8.38376    9.99987 

50 

10 

8.43293     8.43309    9.99984 

50 

20 

8.38450    8.38463    9.99987 

40 

20 

8.  43  371     8.  43  387    9.  99  984 

40 

30 

8.38537    8.38550    9.99987 

30 

30 

8.  43  448    8.  43  464    9.  99  984 

30 

40 

8.38624    8.38636    9.99987 

20 

40 

8.43526    8.43542    9-99984 

20 

50 

8.38710    8.38723    9.99987 

10 

50 

8.43603    8.43619    9.99984 

10 

240 

8.38796    8.38809    9.99987 

036 

340 

8.43680    8.43696    9.99984 

026 

10 

8.  38  882    8.  38  895    9.  99  987 

50 

10 

8.43757  .  8.43773    9.99984 

50 

20 

8.38968    8.38981    9.99987 

40 

20 

8.43834     8.43850    9.99984 

40 

30 

8.39054    8.39067    9.99987 

30 

30 

8.  43  910     8.  43  927    9.  99  984 

30 

40 

8.39139    8.39153    9.99987' 

20 

40 

8.  43  987     8.  44  003     9.  99  984 

20 

50 

8.39225    8.39238    9.99987 

10 

50 

8.44063     8.44080    9.99983 

10 

250 

8.39310    8.39323    9.99987 

035 

350 

8.44139    8.44156    9.99983 

025 

10 

8.  39  395     8.  39  408    9.  99  987 

50 

10 

8.  44  216    8.  44  232    9.  99  983 

50 

20 

8.39480    8.39493     9.99987 

40 

20 

8.  44  292    8.  44  308    9.  99  983 

40 

30 

8.39565    8.39578    9.99987 

30 

30 

8.44367    8.44384    9.99983 

30 

40 

8.  39  649    8.  39  663    9.  99  987 

20 

40 

8.44443     8.44460    9.99983 

20 

50 

8.39734    8.39747    9.99986 

10 

50 

8.44519    8.44536    9.99983 

10 

260 

8.  39  818    8.  39  832    9.  99  986 

034 

360 

8.44594    8.44611     9.99983 

024 

10 

8.  39  902    8.  39  916    9.  99  986 

50 

10 

8.44669    8.44686    9.99983 

50 

20 

8.39986    8.40000    9.99986 

40 

20 

8.44745     8.44762    9.99983 

40 

30 

8.  40  070    8.  40  083     9.  99  986 

30 

30 

8.  44  820    8.  44  837    9.  99  983 

30 

40 

8.  40  153    8.  40  167    9.  99  986 

20 

40 

8.  44  895     8.  44  912    9.  99  983 

20 

50 

8.  40  237    8.  40  251     9.  99  986 

10 

50 

8.44969    8.44987    9.99983 

10 

270 

8.40320    8.40334    9.99986 

033 

370 

8.45044    8.45061     9.99983 

023 

10 

8.40403     8.40417    9.99986 

50 

10 

8.45119    8.45136    9.99983 

50 

20 

8.40486    8.40500    9.99986 

40 

20 

8.  45  193     8.  45  210    9.  99  983 

40 

30 

8.40569    8.40583    9.99986 

30 

30 

8.  45  267    8.  45  285     9.  99  983 

30 

40 

8.40651     8.40665     9.99986 

20 

40 

8.  45  341     8.  45  359    9.  99  982 

20 

50 

8.40734    8.40748    9.99986 

10 

50 

8.  45  415     8.  45  433     9.  99  982 

10 

280 

8.40816    8.40830    9.99986 

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380 

8.  45  489    8.  45  507    9.  99  982 

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8.  40  898    8.  40  913    9.  99  986 

50 

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8.  45  563     8.  45  581     9.  99  982 

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8.  40  980    8.  40  995     9.  99  986 

40 

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8.45637    8.45655    9.99982 

40 

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8.  41  062    8.  41  077    9.  99  986 

30 

30 

8.45710    8.45728    9.99982 

30 

40 

8.41144    8.41158    9.99986 

20 

40 

8.45784    8.45802    9.99982 

20 

50 

8.41225     8.41240    9.99986 

10 

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8.45857    8.45875    9.99982 

10 

290 

8.  41  307    8.  41  321     9.  99  985 

031 

390 

8.45930    8.45948    9.99982 

021 

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8.  41  388    8.  41  403     9.  99  985 

50 

10 

8.  46  003     8.  46  021     9.  99  982 

50 

20 

8.41469    8.41484    9.99985 

40 

20 

8.46076    8.46094    9.99982 

40 

30 

8.41550    8.41565     9.99985 

30 

30 

8.  46  149     8.  46  167    9.  99  982 

30 

40 

8.41631     8.41646    9.99985 

20 

40 

8.46222     8.46240    9.99982 

20 

50 

8.41711    8.41726    9.99985 

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8.46294     8.46312    9.99982 

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8.41792    8.41807    9.99985 

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8.46366    8.46385    9.99982 

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8.  50  504    8.  50  527    9.  99  978 

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40 

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8.50636    8.50658     9.99978 

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8.46583    8.46602    9.99981 

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8.  50  701     8.  50  724    9.  99  978 

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8.46655     8.46674    9.99981 

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8.50897    -8.50920    9.99977 

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8.46870    8.46889    9.99981 

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8.  50  963     8.  50  985    9.  99  977 

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8.46942     8.46960    9.99981 

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8.51092     8.51115     9.99977 

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8.  47  084    8.  47  103    9.  99  981 

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8.51157     8.51180    9.99977 

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8.47155     8.47174    9.99981 

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8.51222     8.51245    9.99977 

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8.  47  226    8.  47  245     9.  99  981 

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8.51287    8.51310    9.99977 

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8.47297    8.47316    9.99981 

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8.51351     8.51374    9.99977 

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8.  47  368    8.  47  387    9.  99  981 

40 

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8.51416    8.51439    9.99977 

40 

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8.47439    8.47458    9.99981 

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8.51480     8.51503     9.99977 

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8.47509    8.47528    9.99981 

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8.51544    8.51568    9.99977 

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8.47580    8.47599    9.99981 

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8.51609    8.51632    9.99977 

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430 

8.  47  650    8.  47  669    9.  99  981    017 

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8.51673     8.51696    9.99977 

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8.47720    8.47740    9.99980 

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8.51737     8.51760    9.99976 

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8.47790    8.47810    9.99980 

40 

20 

8.51801     8.51824    9.99976 

40 

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8.47860    8.47880    9.99980 

30 

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8.51864    8.51888    9.99976 

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8.47930    8.47950    9.99980 

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8.51928    8.51952    9.99976 

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8.48000    8.48020    9.99980 

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8.51992    8.52015     9.99976 

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440 

8.48069    8.48090    Q.  99  980 

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8.52055     8.52079    9.99976 

0    6 

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8.48139    8.48159    9.99980 

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8.52119     8.52143    9.99976 

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8.  48  208    8.  48  228    9.  99  980 

40 

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8.52182     8.52206    9.99976 

40 

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8.  48  278    8.  48  298    9.  99  980 

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8.52245     8.52269    9.99976 

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8.  48  347    8.  48  367    9.  99  980 

20 

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8.52308    8.52332    9.99976 

20 

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8.48416    8.48436    9.99980 

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8.52371     8.52396    9.99976 

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450 

8.  48  485    8.  48  505    9.  99  980 

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8.52434     8.52459    9.99976 

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8.  48  554    8.  48  574    9.  99  980 

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8.52560    8.52584    9.99976 

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8.48691     8.48711     9.99980 

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8.48760    8.48780    9.99979 

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8.48828    8.48849    9.99979 

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log  cos 

log  cot 

JL  vF 

log  tan 

log  sin 

f 

48C 


47° 


43C 


44C 


49 


t 

log  sin 

log  tan 

log  cot 

log  cos 

f 

<) 

C) 

10 

g 

o 

83378 

96966 

03034 

86413 

60 

1 

83392 

96991 

03009 

86401 

59 

2 

83405 

97016 

02984 

86389 

58 

3 

83419 

97042 

02958 

86377 

57 

4 

83432 

97067 

02933 

86366 

56 

5 

83446 

97092 

02908 

86354 

55 

6 

83459 

97118 

02882 

86342 

54 

7 

83473 

97143 

02857 

86330 

53 

8 

83486 

97168 

02832 

86318 

52 

9 

83500 

97193 

02807 

86306 

51 

1O 

83513 

97219 

02781 

86295 

50 

11 

83527 

97244 

02756 

86283 

49 

12 

83540 

97269 

02731 

86271 

48 

13 

83554 

97295 

02705 

86259 

47 

14 

83567 

97320 

02680 

86247 

46 

15 

83581 

97345 

02655 

86235 

45 

16 

83594 

97  371 

02629 

86223 

44 

17 

83608 

97396 

02604 

86211 

43 

18 

83621 

97421 

02579 

86200 

42 

19 

83634 

97447 

02553 

86188 

41 

20 

83648 

97  472 

02528 

86176 

40 

21 

83661 

97497 

02503 

86164 

39 

22 

83674 

97523 

02477 

86152 

38 

23 

83688 

97548 

02452 

86140 

37 

24 

83701 

97573 

02427 

86128 

36 

25 

83715 

97598 

02402 

86116 

35 

26 

83728 

97624 

02  376 

86104 

34 

27 

83741 

97649 

02351 

86092 

33 

28 

83755 

97674 

02326 

86080 

32 

29 

83768 

97700 

02300 

86068 

31 

30 

83781 

97725 

02275 

86056" 

30 

31 

83795 

97750 

02250 

86044 

29 

32 

83808 

97776 

02224 

86032 

28 

33 

83821 

97801 

02199 

86020 

27 

34 

83834 

97826 

02174 

86008 

26 

35 

83848 

97851 

02149 

85996 

25 

36 

83861 

97877 

02123 

85984 

24 

37 

83874 

97902 

02098 

85972 

23 

38 

83887 

97927 

02073 

85960 

22 

39 

83901 

97953 

02  047 

85948 

21 

40 

83914 

97978 

02022 

85936 

2O 

41 

83927 

98003 

01997 

85924 

19 

42 

83940 

98029 

01971 

85912 

18 

43 

83954 

98054 

01946 

85900 

17 

44 

83967 

98079 

01921 

85888 

16 

45 

83980 

98104 

01896 

85876 

15 

46 

83993 

98130 

01870 

85864 

14* 

47 

84006 

98  155 

01845 

85851 

13 

48 

84020 

98180 

01820 

85839 

12 

49 

84033 

98206 

01794 

85827 

11 

50 

84046 

98231 

01769 

85815 

1O 

51 

84059 

98256 

01744 

85803 

-  9 

52 

84072 

98281 

01719 

85791 

8 

53 

84085 

98307 

01693 

85779 

7 

54 

84098 

98332 

01668 

85766 

6 

55 

84112 

98357 

01643 

85754 

5 

56 

84125 

98383 

01617 

85742 

4 

57 

84138 

98408 

01592 

85730 

3 

58 

84151 

98433 

01567 

85718 

2 

59 

84164 

98458 

01542 

85706 

1 

60 

84177 

98484 

01516 

85693 

0 

1  O 

f 

log  cos 

log  cot 

A  VF 

log  tan 

log  sin 

'1 

t 

log  sin 

log  tan 

log  cot 

log  cos 

f 

9 

«) 

10 

9 

0 

84177 

98484 

01516 

85693 

60 

1 

84190 

98509 

01491 

85681 

59 

•  2 

84203 

98534 

01466 

85669 

58 

3 

84216 

98560 

01440 

85657 

57 

4 

84229 

98585 

01415 

85645 

56 

5 

84242 

98610 

01390 

85632 

55 

6 

84255 

98635 

01365 

85620 

54 

7 

84269 

98661 

01339 

85608 

53 

8 

84282 

98686 

01314 

85596 

52 

9 

84295 

98711 

01289 

85583 

51 

1O 

84308 

98737 

01263 

85571 

50 

11 

84321 

98762 

01238 

85559 

49 

12 

84334 

98787 

01213 

85547 

48 

13 

84347 

98812 

01  188 

85534 

47 

14 

84360 

98838 

01162 

85522 

46 

15 

84373 

98863 

01  137 

85  510 

45 

16 

84385 

98888 

01  112 

85497 

44 

17 

84398 

98913 

01087 

85485 

43 

18 

84411 

98939 

01061 

85473 

42 

19 

84424 

98964 

01036 

85460 

41 

20 

84437 

98989 

01011 

85448 

4O 

21 

84450 

99015 

00985 

85436 

39 

22 

84463 

99040 

00960 

85  423 

38 

23 

84476 

99065 

00935 

85411 

37 

24 

84489 

99090 

00910 

85399 

36 

25 

84502 

99116 

00884 

85386 

35 

26 

84515 

99141 

00859 

85374 

34 

27 

84528 

99166 

00834 

85361 

33 

28 

84540 

99191 

00809 

85349 

32 

,29 

84553 

99217 

00783 

85337 

31 

3O 

84  566 

99242 

00758 

85324 

30 

31 

84  579 

99267 

00733 

85312 

29 

32 

84592 

99293 

00707 

85299 

28 

33 

84605 

99318 

00682 

85287 

27 

34 

84618 

99343 

00657 

85274 

26 

35 

84630 

99368 

00632 

85262 

25 

36 

84643 

99394 

00606 

85250 

24 

37 

84656 

99419 

00581 

85237 

23 

38 

84669 

99444 

00556 

85225 

22 

39 

84682 

99469 

00531 

85212 

21 

4O 

84694 

99495 

00505 

85200 

2O 

41 

84707 

99520 

00480 

85  187 

19 

42 

84720 

99545 

00455 

85  175 

18 

43 

84733 

99570 

00430 

85162 

17 

44 

84745 

99596 

00404 

85150 

16 

45 

84758 

99621 

00379 

85137 

15 

46 

84771 

99646 

00354 

85  125 

14 

47 

84784 

99672 

00328 

85  112 

13 

48 

84796 

99697 

00303 

85100 

12 

49 

84809 

99722 

00278 

85087 

11 

50 

84822 

99747 

00253 

85074 

10 

51 

84835 

99773 

00227 

85062 

9 

52 

84847 

99798 

00202 

85049 

8 

53 

84860 

99823 

00177 

85037 

7 

54 

84873 

99848 

00152 

85024 

6 

55 

84885 

99874 

00126 

85012 

5 

56 

84898 

99899 

00101 

84999 

4 

57 

84911 

99924 

00076 

84986 

3 

58 

84923 

99949 

00051 

84974 

2 

59 

84936 

99  975 

00025 

84961 

1 

60 

84949 

00000 

00000 

84949 

O 

i  o 

1  O 

t 

log  cos 

J_  VF 

log  cot 

A  vF 

log  tan 

log  sin 

t 

46C 


45( 


50 


TABLE  IV. 

1      f" 

FOB  DETERMINING  WITH  GREATER 

r 

ACCURACY  THAN  CAN  BE  DONE  BY 

MEANS  OF  TABLE  III.  : 

1. 

log  sin,  log  tan,  and  log  cot,  when 

the  angle  is  between  0°  and  2°  ; 

2. 

log  cos,  log  tan,  and  log  cot,  when 

the  angle  is  between  88°  and  90°  ; 

3. 

The  value  of  the  angle  when  the  logarithm  of  the  function  does  not 

lie  between  the  limits  8. 

54  684  and  11.  45  316. 

• 
FORMULAS   FOE  THE   USE   OF  THE   NUMBERS   S   AND  T. 

I.    When  the  angle  a 

is  between  0°  and  2°  : 

log  sin  a  =  log  a'f  +  S. 

log  o"  =  log  sin  a  —  S, 

log  tan  a  =  log  a"  -f  T. 

=  log  tan  a—  T, 

log  cot  a  =  colog  tan  a. 

=  COlog  COt  a  —  T. 

II.   When  the  angle  a  is  between  88°  and  90°  : 

log  cos  a  =  log  (90°  -  a)"  +  S. 

log  (90°  -  a)"  =  log  COS  a  -  S, 

log  cot  a  =  log  (90°  -  a)"  +  T. 

=  log  COt  a  —  T, 

log  tan  a  =  colog  cot  a. 

=  colog  tan  a—  T, 

<*>X 

and  o  =  90°-  (90°  -a). 

VALUES  OF  S  AND  T, 

a" 

S            log  sin  a 

a"             T            log  tan  a 

a              T             log  tan  a 

0 

__ 

0 

_ 

5  146                     8.  39  713 

4.  68  557 

4.  68  557 

4.  68  567 

2409 

'8.06740 

200 

6.  98  660 

5  424                     8.  41  999 

4.  68  556 

4.  68  558 

4.  68  568 

3417 

8.  21  920 

1726 

7.  92  263 

5  689                      8.  44  072 

4.  68  555 

4.  68  559 

4.  68  569 

3823 

8.  26  795 

2432 

8.  07  156 

5  941                      8.  45  955 

4.  68  555 

4.  68  56.0 

4.  68  570 

4190 

8.  30  776 

2976 

8.  15  924 

6  184                      8.  47  697 

4.  68  554 

4.  68  561 

4.  68  571 

4840 

8.37038 

3434 

8.  22  142 

6417                     8.49305 

4.  68  553 

4.  68  562 

4.  68  572 

5414 

8.  41  904 

3838 

8.  26  973 

6  642                     8.  50  802 

4.  68  552 

4.  68  563 

4.  68  573 

5932 

8.  45  872 

4204 

8.30930 

6859                     8.52200 

4.68551 

4.  68  564 

4.  68  574 

6408 

8.  49  223 

4540 

8.  34  270 

7  070                     8.  53  516 

4.  68  550 

4.  68  565 

4.  68  575 

6633 

8.  50  721 

4699 

8.  35  766 

7  173                      8.  54  145 

4.  68  550 

4.  68  565 

4.  68  575 

6851 

8.  52  125 

4853 

8.  37  167 

7  274                     8.  54  753 

4.  68  549 

4.  68  566 

7267 

8.54684 

5146 

8.  39  713 

a" 

8            log  sin  a 

a"             T            log  tan  a 

a             T            log  tan  a 

TABLE  V.— CIRCUMFERENCES  AND  AREAS  OF  CIRCLES. 


If  N  =  the  radius  of  the  circle,  the  circumference  =  2  Tr-ZV. 

If  N  =  the  radius  of  the  circle,  the  area                  =  irN2. 

If  N  =  the  circumference  of  the  circle,  the  radius  =  —  N. 

IfN=  the  circumference  of  the  circle,  the  area     =  —  N'2. 

47T 

N 

27TJV                 IT}?*                JLjVT           J_^2 
27T                   47T 

N 

*™     **•    &*   £*• 

O 

0.00            0.0     0.000         0.00 

50 

314.16       7854       7.96     198.94 

1 

6.  28            3.  1      0.  159         0.  08 

51 

320.44        8171        8.12      206.98 

2 

12.57           12.6     0.318          0.32 

52 

326.73        8495        8.28      215.18 

3 

18.85          28.3      0.477         0.72 

53 

333.01        8825        8.44      223.53 

4 

25.13          50.3      0.637          1.27 

54 

339.29        9161        8.59      232.05 

5 

31.42          78.5      0.796         ].99 

55 

345.58       9503       8.75      240.72 

6 

37.70        113.1      0.955          2.86 

56 

351.86       9852       8.91      249.55 

7 

43.98        153.9      1.114         3.90 

57 

358.14      10207       9.07'    258.55 

8 

50.27        201.1      1.273          5.09 

58 

364.42      10568       9.23      267.70 

9 

56.55        254.5      1.432         6.45 

59 

370.71      10936       9.39      277.01 

1O 

62.83        314.2      1.592         7.96 

60 

376.99      11310       9.55      286.48 

11 

69.12        380.1      1.751         9.63 

61 

383.27      11690       9.71      296.11 

12 

75.40        452.4      1.910       11.46 

62 

389.56      12076       9.87     305.90 

13 

81.68        530.9     2.069        13.45 

63 

395.84      12469      10.03      315.84 

14 

87.96        615.8      2.228        15.60 

64 

402.12      12868      10.19      325.95 

15 

94.25         706.9      2.387        17.90 

65 

408.41      13273      10.35      336.21 

16 

100.53         804.2      2.546        20.37 

66 

414.69      13685      10.50     346.64 

17 

106.81         907.9      2.706        23.00 

67 

420.97      14103      10.66     357.22 

18 

113.10      1017.9     2.865        25.78 

68 

427.26      14527      10.82     367.97 

19 

119.38      1134.1      3.024        28.73 

69 

433.54      14957      10.98     378.87 

20 

125.66      1256.6     3.183       31.83 

70 

439.82      15394      11.14     389.93 

21 

131.95      1385.4      3.342        35.09 

71 

446.11      15837      11.30     401.15 

22 

138.23      1520.5      3.501        38.52 

72 

452.39      16286      11.46     412.53 

23 

14*4.51      1661.9      3.661        42.10 

73 

458.67      16742      11.62      424.07 

24 

150.80      1809.6     3.820       45.84 

74 

464.96      17203      11.78     435.77 

25 

157.08      1963.5      3.979        49.74 

75 

471.24      17671      11.94     447.62 

26 

163.  36      2  123.  7      4.  138        53.  79 

76 

477.52      18146      12.10     459.64 

27 

169.65      2290.2      4.297        58.01 

77 

483.81      18627      12.25      471.81 

28 

175.93      2463.0      4.456        62.39 

78 

490.09      19113      12.41      484.15 

29 

182.21      2642.1      4.615        66.92 

79 

496.37      19607      12.57     496.64 

30 

188.50      2827.4     4.775        71.62 

8O 

502.65      20106      12.'  73      509.30 

31 

194.  78      3  019.  1      4.  934        76.  47 

81 

508.94      20612      12.89      522.11 

32 

201.06     3217.0      5.093        81.49 

82 

515.22      21124      13.05     535.08 

33 

207.35      3421.2      5.252        86.66 

83 

521.50      21642      13.21      548.21 

34 

213.63      3631.7      5.411        91.99 

84 

527.79      22167      13.37      561.50 

35 

219.91      3848.5      5.570       97.48 

85 

534.07      22698      13.53      574.95 

36 

226.19      4071.5      5.730      103.13 

86 

540.35      23235      13.69     588.55 

37 

232.48      4300.8      5.889      108.94 

87 

546.64      23779      13.85      602.32 

38 

238.  76      4  536.  5      6.  048      114.  91 

88 

552.92      24328      14.01      616.25 

39 

245.04      4778.4      6.207      121.04 

89 

559.20      24885      14.16     630.33 

40 

251.33      5026.5      6.366      127.32 

9O 

565.49      25447      14.32     644.58 

41 

257.61      5281.0     6.525      133.77 

91 

571.77      26016      14.48      658.98 

42 

263.89     5541.8      6.685      140.37 

92 

578.05      26590      14.64     673.54 

43 

270.  18      5  808.  8      6.  844      147.  14 

93 

584.34      27172      14.80      688.27 

44 

276.  46     6  082.  1      7.  003      154.  06 

94 

590.62      27759      14.96      703.15 

45 

282.74      6361.7      7.162      161.14 

95 

596.90      28353      1512      718.19 

46 

289.03      6647.6      7.321      168.39 

96 

603.19      28953      15,28      733.39 

47 

295.31      6939.8      7.480      175.79 

97 

609.47      29559      15.44      748.74 

48 

301.59      7238.2      7.639      183.35 

98 

615.75      30172      15.60      764.26 

49 

307.88      7543.0      7.799      191.07 

99 

622.04     30791      15.76      779.94 

50 

314.46      7854.0      7.958      198.94 

1OO 

628.32     31416      15.92      795.77 

* 

2ir             4ir 

N 

27T                 4K 

52 


TABLE  VI. -NATFEAL  FUNCTIONS. 


/ 

O° 

1° 

2o 

3° 

4° 

r 

sin   cos 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

o 

0000  1.000 

0175  9998 

0349  9994 

0523  9986 

0698  9976 

6O 

1 

0003  1.000 

0177  9998 

0352  9994 

0526  9986 

0700  9975 

59 

2 

0006  1.000 

0180  9998 

0355  9994 

0529  9986 

0703  9975 

58 

3 

0009  1.000 

0183  9998 

0358  9994 

0532  9986 

0706  9975 

57 

4 

0012  1.000 

0186  9998 

0361  9993 

0535  9986 

0709  9975 

56 

5 

0015  1.000 

0189  9998 

0364  9993 

0538  9986 

0712  9975 

55 

6 

0017  1.000 

0192  9998 

0366  9993 

0541  9985 

0715  9974 

54 

7 

0020  1.000 

0195  9998 

0369  9993 

0544  9985 

0718  9974 

53 

8 

0023  1.000 

0198  9998 

0372  9993 

0547  9985 

0721  9974 

52 

9 

,  0026  1.000 

0201  9998 

0375  9993 

0550  9985 

0724  9974 

51 

1O 

0029  1.000 

0204  9998 

0378  9993 

0552  9985 

0727  9974 

50 

11 

0032  1.000 

0207  9998 

0381  9993 

0555  9985 

0729  9973 

49 

12 

0035  1.000 

0209  9998 

0384  9993 

0558  9984 

0732  9973 

48 

13 

0038  1.000 

0212  9998 

0387  9993 

0561  9984 

0735  9973 

47 

14 

0041  1.000 

0215  9998 

0390  9992 

0564  9984 

0738  9973 

46 

15 

0044  1.000 

0218  9998 

0393  9992 

0567  9984 

0741  9973 

45 

16 

0047  1.000 

0221  9998 

0396  9992 

0570  9984 

0744  9972 

44 

17 

0049  1.000 

0224  9997 

0398  9992 

0573  9984 

0747  9972 

43 

18 

0052  1.000 

0227  9997 

0401  9992 

0576  9983 

0750  9972 

42 

19 

0055  1.000 

0230  9997 

0404  9992 

0579  9983 

0753  9972 

41 

2O 

0058  1.000 

0233  9997 

0407  9992 

0581  9983 

0756  9971 

4O 

21 

0061  1.000 

0236  9997 

0410  9992 

0584  9983 

0758  9971 

39 

22 

0064  1.000 

0239  9997 

0413  9991 

0587  9983 

0761  9971 

38 

23 

0067  1.000 

0241  9997 

0416  9991 

0590  9983 

0764  9971 

37 

24 

0070  1.000 

0244  9997 

0419  9991 

0593  9982 

0767  9971 

36 

25 

0073  1.000 

0247  9997 

0422  9991 

0596  9982 

0770  9970 

35 

26 

0076  1.000 

0250  9997 

0425  9991 

0599  9982 

0773  9970 

34 

27 

0079  1.000 

0253  9997 

0427  9991 

0602  9982 

0776  9970 

33 

28 

0081  1.000 

0256  9997 

0430  9991 

0605  9982 

0779  9970 

32 

29 

0084  1.000 

0259  9997 

0433  9991 

0608  9982 

0782  9969 

31 

3O 

0087  1.000 

0262  9997 

0436  9990 

0610  9981 

0785  9969 

3O 

31 

0090  1.000 

0265  9996 

0439  9990 

0613  9981 

0787  9969 

29 

32 

0093  1.000 

0268  9996 

0442  9990 

0616  9981 

0790  9969* 

28 

33 

0096  1.000 

0270  9996 

0445  9990 

0619  9981 

0793  9968 

27 

34 

0099  1.000 

0273  9996 

0448  9990 

0622  9981 

0796  9968 

26 

35 

0102  9999 

0276  9996 

0451  9990 

0625  9980 

0799  9968 

25 

36 

0105  9999 

0279  9996 

0454  9990 

0628  9980 

0802  9968 

24 

37 

0108  9999 

0282  9996 

0457  9990 

0631  9980 

0805  9968 

23 

38 

0111  9999 

0285  9996 

0459  9989 

0634  9980 

0808  9967 

22 

39 

0113  9999 

0288  9996 

0462  9989 

0637  9980 

0811  9967 

21 

40 

0116  9999 

0291  9996 

0465  9989 

0640  9980 

0814  9967 

20 

41 

0119  9999 

0294  9996 

0468  9989 

0642  9979 

0816  9967 

19 

42 

0122  9999 

0297  9996 

0471  9989 

0645  9979 

0819  9966 

18 

43 

0125  9999 

0300  9996 

0474  9989 

0648  9979 

0822  9966 

17 

44 

0128  9999 

0302  9995 

0477  9989 

0651  9979 

0825  9966 

16 

45 

0131  9999 

0305  9995 

0480  9988 

0654  9979 

0828  9966 

15 

46 

0134  9999 

0308  9995 

0483  9988 

0657  9978 

0831  9965 

14 

47 

0137  9999 

0311  9995 

0486  9988 

0660  9978 

0834  9965 

13 

48 

0140  9999 

0314  9995 

0488  9988 

0663  9978 

0837  9965 

.12 

49 

0143  9999 

0317  9995 

0491  9988 

0666  9978 

0840  9965 

11 

5O 

0145  9999 

0320  9995 

0494  9988 

0669  9978 

0843  9964 

1O 

51 

0148  9999 

0323  9995 

0497  9988 

0671  9977 

0845  9964 

9 

52 

0151  9999 

0326  9995 

0500  9987 

0674  9977 

0848  9964 

8 

53 

0154  9999 

0329  9995 

0503  9987 

0677  9977 

0851  9964 

7 

54 

0157  9999 

0332  9995 

0506  9987 

0680  9977 

0854  9963 

6 

55 

0160  9999 

0334  9994 

0509  9987 

0683  9977 

0857  9963 

5 

56 

0163  9999 

0337  9994 

0512  9987 

0686  9976 

0860  9963 

4 

57 

0166  9999 

0340  9994 

0515  9987 

0689  9976 

0863  9963 

3 

58 

0169  9999 

0343  9994 

0518  9987 

0692  9976 

0866  9962 

2 

59 

0172  9999 

0346  9994 

0520  9986 

0695  9976 

0869  9962 

1 

6O 

0175  9999 

0349  9994 

0523  9986 

0698  9976 

0872  9962 

0 

cos   sin 

cos  sin 

'cos  sin 

cos  sin 

cos   sin 

f 

89° 

88° 

87° 

86° 

85° 

r 

NATURAL   SINES   AND   COSINES. 


58 


f 

5° 

6° 

7° 

8° 

9° 

f 

sin   cos 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

o 

0872  9962 

1045  9945 

1219  9925 

1392  9903 

1564  9877 

60 

1 

0874  9962 

1048  9945 

1222  9925 

1395  9902 

1567  9876 

59 

2 

0877  9961 

1051  9945 

1224  9925 

1397  9902 

1570  9876 

58 

3 

0880  9961 

1054  9944 

1227  9924 

1400  9901 

1573  9876 

57 

4 

0883  9961 

1057  9944 

1230  9924 

1403  9901 

1576  9875 

56 

5 

0886  9961 

1060  9944 

1233  9924 

1406  9901 

1579  9875 

55 

6 

0889  9960 

1063  9943 

1236  9923 

1409  9900 

1582  9874 

54 

7 

0892  9960 

1066  9943 

1239  9923 

1412  9900 

1584  9874 

53 

8 

0895  9960 

1068  9943 

1241  9923 

1415  9899 

1587  9873 

52 

9 

0898  9960 

1071  9942 

1245  9922 

1418  9899 

1590  9873 

51 

1C 

0901  9959 

1074  9942 

1248  9922 

1421  9899 

1593  9872 

50 

11 

0903  9959 

1077  9942 

1250  9922 

1423  9898 

1596  9872 

49 

12 

0906  9959 

1080  9942 

1253  9921 

1426  9898 

1599  9871 

48 

13 

0909  9959 

1083  9941 

1256  9921 

1429  9897 

1602  9871 

47 

14 

0912  9958 

1086  9941 

1259  9920 

1432  9897 

1605  9870 

46 

15 

0915  9958 

1089  9941 

1262  9920 

1435  9897 

1607  9870 

45 

16 

0918  9958 

1092  9940 

1265  9920 

1438  9896 

1610  9869 

44 

17 

0921  9958 

1094  9940 

1268  9919 

1441  9896 

1613  9869 

43 

18 

0924  9957 

1097  9940 

1271  9919 

1444  9895 

1616  9869 

42 

19 

0927  9957 

1100  9939 

1274  9919 

1446  9895 

1619  9868 

41 

2O 

0929  9957 

1103  9939 

1276  9918 

1449  9894 

1622  9868 

40 

21 

0932  9956 

1106  9939 

1279  9918 

1452  9894 

1625  9867 

39 

22 

0935  9956 

1109  9938 

1282  9917 

1455  9894 

1628  9867 

38 

23 

0938  9956 

1112  9938 

1285  9917 

1458  9893 

1630  9866 

37 

24 

0941  9956 

1115  9938 

1288  9917 

1461  9893 

1633  9866 

36 

25 

0944  9955 

1118  9937 

1291  9916 

1464  9892 

1636  9865 

35 

26 

0947  9955 

1120  9937 

1294  9916 

1467  9892 

1639  9865 

34 

27 

0950  9955 

1123  9937 

1297  9916 

1469  9891 

1642  9864 

33 

28 

0953  9955 

1126  9936 

1299  9915 

1472  9891 

1645  9864 

32 

29 

0956  9954 

1129  9936 

1302  9915 

1475  9891 

1648  9863 

31 

30 

0958  9954 

1132  9936 

1305  9914 

1478  9890 

1650  9863 

30 

31 

0961  9954 

1135  9935 

1308  9914 

1481  9890 

1653  9862 

29 

32 

0964  9953 

1138  9935 

1311  9914 

1484  9889 

1656  9862 

28 

33 

0967  9953 

1141  9935 

1314  9913 

1487  9889 

1659  9861 

27 

34 

0970  9953 

1144  9934 

1317  9913 

1490  9888 

1662  9861 

26 

35 

0973  9953 

1146  9934 

1320  9913 

1492  9888 

1665  9860 

25 

36 

0976  9952 

1149  9934 

1323  9912 

1495  9888 

1668  9860 

24 

37 

0979  9952 

1152  9933 

1325  9912 

1498  9887 

1671  9859 

23 

38 

0982  9952 

1155  9933 

1328  9911 

1501  9887 

1673  9859 

22 

39 

0985  9951 

1158  9933 

1331  9911 

1504  9886 

1676  9859 

21 

40 

0987  9951 

1161  9932 

1334  9911 

1507  9886 

1679  9858 

20 

41 

0990  9951 

1164  9932 

1337  9910 

1510  9885 

1682  9858 

19 

42 

0993  9951 

1167  9932 

1340  9910 

1513  9885 

1685  9857 

18 

43 

0996  9950  - 

1170  £931 

1343  9909 

1515  9884 

1688  9857 

17 

44 

0999  9950 

1172  9931 

1346  9909 

1518  9884 

1691  9856 

16 

45 

1002  9950 

1175  9931 

1349  9909 

1521  9884 

1693  9856 

15 

46 

1005  9949 

1178  9930 

1351  9908 

1524  9883 

1696  9855 

14 

47 

1008  9949 

1181  9930 

1354  9908 

1527  9883 

1699  9855 

13 

48 

1011  9949 

1184  9930 

1357  9907 

1530  9882 

1702  9854 

12 

49 

1013  9949 

1187  9929 

1360  9907 

1533  9882 

1705  9854 

11 

50 

1016  9948 

1190  9929 

1363  9907 

1536  9881 

1708  9853 

1O 

51 

1019  9948 

1193  9929 

1366  9906 

1538  9881 

1711  9853 

9 

52 

1022  9948 

1196  9928 

1369  9906 

1541  9880 

1714  9852 

8 

53 

1025  9947 

1198  9928 

1372  9905 

1544  9880 

1716  9852 

7 

54 

1028  9947 

1201  9928 

1374  9905 

1547  9880 

1719  9851 

6 

55 

1031  9947 

1204  9927 

1377  9905 

1550  9879 

1722  9851 

5 

56 

1034  9946 

1207  9927 

1380  9904 

1553  9879 

1725  9850 

4 

57 

1037  9946 

1210  9927 

1383  9904 

1556  9878 

1728  9850 

3 

58 

1039  9946 

1213  9926 

1386  9903 

1559  9878 

1731  9849 

2 

59 

1042  9946 

1216  9926 

1389  9903 

1561  9877 

1734  9849 

1 

60 

1045  9945 

1219  9925 

1392  9903 

1564  9877 

1736  9848 

0 

cos  sin 

cos  sin 

cos  sin 

cos  sin 

cos  sin 

'      84° 

83° 

82° 

81° 

80° 

t 

54 


NATURAL   SINES   AND   COSINES. 


/ 

10° 

11° 

12° 

13° 

14° 

r 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

0 

1736  9848 

1908  9816 

2079  9781 

2250  9744 

2419  9703 

6O 

1 

1739  9848 

1911  9816 

2082  9781 

2252  9743 

2422  9702 

59 

2 

1742  9847 

1914  9815 

2085  9780 

2255  9742 

2425  9702 

58 

3 

1745  9847 

1917  9815 

2088  9780 

2258  9742 

2428  9701 

57 

4 

1748  9846 

1920  9814 

2090  9779 

2261  9741 

2431  9700 

56 

5 

1751  9846 

1922  9813 

2093  9778 

2264  9740 

2433  9699 

55 

6 

1754  9845 

1925  9813 

2096  9778 

2267  9740 

2436  9699 

54 

7 

1757  9845 

1928  9812 

2099  9777 

2269  9739 

2439  9698 

53 

8 

1759  9844 

1931  9812 

2102  9777 

2272  9738 

2442  9697 

52 

9 

1762  9843 

1934  9811 

2105  9776 

2275  9738 

2445  9697 

51 

10 

1765  9843 

1937  9811 

2108  9775 

2278  9737 

2447  9696 

50 

11 

1768  9842 

1939  9810 

2110  9775 

2281  9736 

2450  9695 

49 

12 

1771  9842 

1942  9810 

2113  9774 

2284  9736 

2453  9694 

48 

13 

1774  9841 

1945  9809 

2116  9774 

2286  9735 

2456  9694 

47 

14 

1777  9841 

1948  9808 

2119  9773 

2289  9734 

2459  9693 

46 

15 

1779  9840 

1951  9808 

2122  9772 

2292  9734 

2462  9692 

45 

16 

1782  9840 

1954  9807 

2125  9772 

2295  9733 

2464  9692 

44 

17 

1785  9839 

1957  9807 

2127  9771 

2298  9732 

2467  9691 

43 

18 

1788  9839 

1959  9806 

2130  9770 

2300  9732 

2470  9690 

42 

19 

1791  9838 

1962  9806 

2133  9770 

2303  9731 

2473  9689 

41 

20 

1794  9838 

1965  9805 

2136  9769 

2306  9730 

2476  9689 

4O 

21 

1797  9837 

1968  9804 

2139  9769 

2309  9730 

2478  9688 

39 

22 

1799  9837 

1971  9804 

2142  9768 

2312  9729 

2481  9687 

38 

23 

1802  9836 

1974  9803 

2145  9767 

2315  9728 

2484  9687 

37 

24 

1805  9836 

1977  9803 

2147  9767 

2317  9728 

2487  9686 

36 

25 

1808  9835 

1979  9802 

2150  9766 

2320  9727 

2490  9685 

35 

26 

1811  9835 

1982  9802 

2153  9765 

2323  9726 

•   2493  9684 

34 

27 

1814  9834 

1985  9801 

2156  9765 

2326  9726 

2495  9684 

33 

28 

1817  9834 

1988  9800 

2159  9764 

2329  9725 

2498  9683 

32 

29 

1819  9833 

1991  9800 

2162  9764 

2332  9724 

2501  9682 

31 

30 

1822  9833 

1994  9799 

2164  9763 

2334  9724 

2504  9681 

30 

31 

1825  9832 

1997  9799 

2167  9762 

2337  9723 

2507  9681 

29 

32 

1828  9831 

1999  9798 

2170  9762 

2340  9722 

2509  9680 

28 

33 

1831  9831 

2002  9798 

2173  9761 

2343  9722 

2512  9679 

27 

34 

1834  9830 

2005  9797 

2176  9760 

2346  9721 

2515  9679 

26 

35 

1837  9830 

2008  9796 

2179  9760 

2349  9720 

2518  9678 

25 

36 

1840  9829 

2011  9796 

2181  9759 

2351  9720 

2521  9677 

24 

37 

1842  9829 

2014  9795 

2184  9759 

2354  9719 

2524  9676 

23 

38 

1845  9828 

2016  9795 

2187  9758 

2357  9718 

2526  9676 

22 

39 

1848  9828 

2019  9794 

2190  9757 

2360  9718 

2529  9675 

21 

40 

1851  9827 

2022  9793 

2193  9757 

2363  9717 

2532  9674 

2O 

41 

1854  9827 

2025  9793 

2196  9756 

2366  9716 

2535  9673 

19 

42 

1857  9826 

2028  9792 

2198  9755 

2368  9715 

2538  9673 

18 

43 

1860  9826 

2031  9792 

2201  9755 

2371  9715 

.  2540  9672 

17 

44 

1862  9825 

2034  9791 

2204  9754 

2374  9714 

2543  9671 

16 

45 

1865  9825 

2036  9790 

2207  9753 

2377  9713 

2546  9670 

15 

46 

1868  9824 

2039  9790 

2210  9753 

2380  9713 

2549  9670 

14 

47 

1871  9823 

2042  9789 

2213  9752 

2383  9712 

2552  9669 

13 

48 

1874  9823 

2045  9789 

2215  9751 

2385  9711 

2554  9668 

12 

49 

1877  9822 

2048  9788 

2218  9751 

2388  9711 

2557  9667 

11 

50 

1880  9822 

2051  9787 

2221  9750 

2391  9710 

2560  9667 

10 

51 

1882  9821 

2054  9787 

2224  9750 

2394  9709 

2563  9666 

9 

52 

1885  9821 

2056  9786 

2227  9749 

2397  9709 

2566  9665 

8 

53 

1888  9820 

2059  9786 

2230  9748 

2399  9708 

2569  9665 

7 

54 

]891  9820 

2062  9785 

2233  9748 

2402  9707 

2571  9664 

6 

55 

1894  9819 

2065  9784 

2235  9747 

2405  9706 

2574  9663 

5 

56 

1897  9818 

2068  9784 

2238  9746 

2408  9706 

2577  9662 

4 

57 

1900  9818 

2071  9783 

2241  9746 

2411  9705 

2580  9662 

3 

58 

1902  9817 

2073  9783 

2244  9745 

2414  9704 

2583  9661 

2 

59 

1905  9817 

2076  9782 

2247  9744 

2416  9704 

2585  9660 

1 

6O 

1908  9816 

2079  9781 

2250  9744 

2419  9703 

2588  9659 

0 

cos  sin 

cos  sin 

cos  sin 

cos  sin 

cos   sin 

f 

79° 

78° 

77° 

76° 

75° 

f 

NATURAL    SINES   AND   COSINES. 


55 


/ 

15° 

16° 

17° 

18° 

19° 

f 

sin  cos 

siii  cos 

sin  cos 

sin  cos 

sin  cos 

o 

2588  9659 

2756  9613 

2924  9563 

3090  9511 

3256  9455 

60 

1 

2591  9659 

2759  9612 

2926  9562 

3093  9510 

3258  9454 

59 

2 

2594  9658 

2762  9611 

2929  9561 

3096  9509 

3261  9453 

58 

3 

2597  9657 

2765  9610 

2932  9560 

3098  9508 

3264  9452 

57 

4 

2599  9656 

2768  9609 

2935  9560 

3101  9507 

3267  9451 

56 

5 

2602  9655 

2770  9609 

2938  9559 

3104  9506 

3269  9450 

55 

6 

2605  9655 

2773  9608 

2940  9558 

3107  9505 

3272  9449 

54 

7 

2608  9654 

2776  9607 

2943  9557 

3110  9504 

3275  9449 

53 

8 

2611  9653 

2779  9606 

2<H6  9556 

3112  9503 

3278  9448 

52 

9 

2613  9652 

2782  9605 

2949  9555 

3115  9502 

3280  9447 

51 

10 

2616  9652 

2784  9605 

2952  9555 

3118  9502 

3283  9446 

50 

11 

2619  9651 

2787  9604 

2954  9554 

3121  9501 

3286  9445 

49 

12 

2622  9650 

2790  9603 

2957  9553 

3123  9500 

3289  9444 

48 

13 

2625  9649 

2793  9602 

2960  9552 

3126  9499 

3291  9443 

47 

14 

2628  9649 

2795  9601 

2963  9551 

3129  9498 

3294  9442 

46 

15 

2630  9648 

2798  9600 

2965  9550 

3132  9497 

3297  9441 

45 

16 

2633  9647 

2801  9600 

2968  9549 

3134  9496 

3300  9440 

44 

17 

2636  9646 

2804  9599 

2971  9548 

3137  9495 

3302  9439 

43 

18 

2639  9646 

2807  9598 

2974  9548 

3140  9494 

3305  9438 

42 

19 

2642  9645 

2809  9597 

2977  9547 

3143  9493 

3308  9437 

41 

20 

2644  9644 

2812  9596 

2979  9546 

3145  9492 

3311  9436 

4O 

21 

2647  9643 

2815  9596 

2982  9545 

3148  9492 

3313  9435 

39 

22 

2650  9642 

2818  9595 

2985  9544 

3151  9491 

3316  9434 

38 

23 

2653  9642 

2821  9594 

2988  9543 

3154  9490 

3319  9433 

37 

24 

2656  9641 

2823  9593 

2990  9542 

3156  9489 

3322  9432 

36 

25 

2658  9640 

2826  9592 

2993  9542 

3159  9488 

3324  9431 

35 

26 

2661  9639 

2829  9591 

2996  9541 

3162  9487 

3327  9430 

34 

27 

2664  9639 

2832  9591 

2999  9540 

3165  9486 

3330  9429 

33 

28 

2667  9638 

2835  9590 

3002  9539 

3168  9485 

3333  9428 

32 

29 

2670  9637 

2837  9589 

3004  9538 

3170  9484 

3335  9427 

31 

30 

2672  9636 

2840  9588 

3007  9537 

3173  9483 

3338  9426 

30 

31 

2675  9636 

2843  9587 

3010  9536 

3176  9482 

3341  9425 

29 

32 

2678  9635 

2846  9587 

3013  9535 

3179  9481 

3344  9424 

28 

33 

2681  9634 

2849  9586 

3015  9535 

3181  9480 

3346  9423 

27 

34 

2684  9633 

2851  9585 

3018  9534 

3184  9480 

3349  9423 

26 

35 

2686  9632 

2854  9584 

3021  9533 

3187  9479 

3352  9422 

25 

36 

2689  9632 

2857  9583 

3024  9532 

3190  9478 

3355  9421 

24 

37 

2692  9631 

2860  9582 

3026  9531 

3192  9477 

3357  9420 

23 

38 

2695  9630 

2862  9582 

3029  9530 

3195  9476 

3360  9419 

22 

39 

2698  9629 

2865  9581 

3032  9529 

3198  9475 

3363  9418 

21 

4O 

2700  9628 

2868  9580 

3035  9528 

3201  9474 

3365  9417 

2O 

41 

2703  9628 

2871  9579 

3038  9527 

3203  9473 

3368  9416 

19 

42 

2706  9627 

2874  9578 

3040  9527 

3206  9472 

3371  9415 

18 

43 

2709  9626 

2876  9577 

3043  9526 

3209  9471 

3371  9414 

17 

44 

2712  9625 

2879  9577 

3046  9525 

3212  9470 

3376  9413 

16 

45 

2714  9625 

2882  9576 

3049  9524 

3214  9469 

3379  9412 

15 

46 

2717  9624 

2885  9575 

3051  9523 

3217  9468 

3382  9411 

14 

47 

2720  9623 

2888  9574 

3054  9522 

3220  9467 

3385  9410 

13 

48 

2723  9622 

2890  9573 

3057  9521 

3223  9466 

3387  9409 

12 

49 

2726  9621 

2893  9572 

3060  9520 

3225  9466 

3390  9408 

11 

5O 

2728  9621 

2896  9572 

3062  9520 

3228  9465 

3393  9407 

1O 

51 

2731  9620 

2899  .9571 

3065  9519 

3231  9464 

3396  9406 

9 

52 

2734  9619 

2901  9570 

3068  9518 

3234  9463 

3398  9405 

8 

53 

2737  9618 

2904  9569 

3071  9517 

3236  9462 

3401  9404 

7 

54 

2740  9617 

2907  9568 

3074  9516 

3239  9461 

3404  9403 

6 

55 

2742.  9617 

2910  9567 

3076  9515 

3242  9460 

3407  9402 

5 

56 

2745  9616 

2913  9566 

3079  9514 

3245  9459 

3409  9401 

4 

57 

2748  9615 

2915  9566 

3082  9513 

3247  9458 

3412  9400 

3 

58 

2751  9614 

2918  9565 

3085  9512 

3250  9457 

3415  9399 

2 

59 

2754  9613 

2921  9564 

3087  9511 

3253  9456 

3417  9398 

1 

6O 

2756  9613 

2924  9563 

3090  9511 

3256  9455 

3420  9397 

0 

cos  sin 

cos  sin 

cos  sin 

cos  sin 

cos   sin 

t 

74° 

73° 

72° 

71° 

70° 

f 

56 


NATURAL   SINES    AND    COSINES. 


/ 

2O° 

21° 

22° 

23° 

24° 

f 

sin   cos 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

0 

3420  9397 

3584  9336 

3746  9272 

3907  9205 

4067  9135 

60 

1 

3423  9396 

3586  9335 

3749  9271 

3910  9204 

4070  9134 

59 

2 

3426  9395 

3589  9334 

3751  9270 

3913  9203 

4073  9133 

58 

3 

3428  9394 

3592  9333 

3754  9269 

3915  9202 

4075  9132 

57 

4 

3431  9393 

3595  9332 

3757  9267 

3918  9200 

4078  9131 

56 

5 

3434  9392 

3597  9331 

3760  9266 

3921  9199 

4081  9130 

55 

6 

3437  9391 

3600  9330 

3762  9265 

3923  9198 

4083  9128 

54 

7 

3439  9390 

3603  9328 

3765  9264 

3926  9197 

4086  9127 

53 

8 

3442  9389 

3605  9327 

3768  9263 

3929  9196 

4089  9126 

52 

9 

3445  9388 

3608  9326 

3770  9262 

3931  9195 

4091  9125 

51 

1O 

3448  9387 

3611  9325 

3773  9261 

3934  9194 

4094  9124 

5O 

11 

3450  9386 

3614  9324 

3776  9260 

3937  9192 

4097  9122 

49 

12 

3453  9385 

3616  9323 

3778  9259 

3939  9191 

4099  9121 

48 

13 

3456  9384 

3619  9322 

3781  9258 

3942  9190 

4102  9120 

47 

14 

3458  9383 

3622  9321 

3784  9257 

3945  9189 

4105  9119 

46 

15 

3461  9382 

3624  9320 

3786  9255 

3947  9188 

4107  9118 

45 

16 

3464  9381 

3627  9319 

3789  9254 

3950  9187 

4110  9116 

44 

17 

3467  9380 

3630  9318 

3792  9253 

3953  9186 

4112  9115 

43 

18 

3469  9379 

3633  9317 

3795  9252 

3955  9184 

4115  9114 

42 

19 

3472  9378 

3635  9316 

3797  9251 

3958  9183 

4118  9113 

41 

2O 

3475  9377 

3638  9315 

3800  9250 

3961  9182 

4120  9112 

40 

21 

3478  9376 

3641  9314 

3803  9249 

3963  9181 

4123  9110 

39 

22 

3480  9375 

3643  9313 

3805  9248 

3966  9180 

4126  9109 

38 

23 

3483  9374 

3646  9312 

3808  9247 

3969  9179 

4128  9108 

37 

24 

3486  9373 

3649  9311 

3811  9245 

3971  9178 

4131  9107 

36 

25 

3488  9372 

3651  9309 

3813  9244 

3974  9176 

4134  9106 

35 

26 

3491  9371 

3654  9308 

3816  9243 

3977  9175 

4136  9104 

34 

27 

3494  9370 

3657  9307 

3819  9242 

3979  9174 

4139  9103 

33 

28 

3497  9369 

3660  9306 

3821  9241 

3982  9173 

4142  9102 

32 

29 

3499  9368 

3662  9305 

3824  9240 

3985  9172 

4144  9101 

31 

30 

3502  9367 

3665  9304 

3827  9239 

3987  9171 

4147  9100 

30 

31 

3505  9366 

3668  9303 

3830  9238 

3990  9169 

4150  9098 

29 

32 

3508  9365 

3670  9302 

3832  9237 

3993  9168 

4152  9097 

28 

33 

3510  9364 

3673  9301 

3835  9235 

3995  9167 

4155  9096 

27 

34 

3513  9363 

3676  9300 

3838  9234 

3998  9166 

4158  9095 

26 

35 

3516  9362 

3679  9299 

3840  9233 

4001  9165 

4160  9094 

25 

36 

3518  9361 

3681  9298 

3843  9232 

4003  9164 

4163  9092 

24 

37 

3521  9360 

3684  9297 

3846  9231 

4006  9162 

4165  9091 

23 

38 

3524  9359 

3687  9296 

3848  9230 

4009  9161 

4168  9090 

22 

39 

3527  9358 

3689  9295 

3851  9229 

4011  9160 

4171  9088 

21 

4O 

3529  9356 

3692  9293 

3854  9228 

4014  9159 

4173  9088 

2O 

41 

3532  9355 

3695  9292 

3856  9227 

4017  9158 

4176  9086 

19 

42 

3535  9354 

3697  9291 

3859  9225 

4019  9157 

4179  9085 

18 

43 

3537  9353 

3700  9290 

3862  9224 

4022  9155 

4181  9084 

17 

44 

3540  9352 

3703  9289 

3864  9223 

4025  9154 

4184  9083 

16 

45 

3543  9351 

3706  9288 

3867  9222 

4027  9153 

4187  9081 

15 

46 

3546  9350 

3708  9287 

3870  9221 

4030  9152 

4189  9080 

14 

47 

3548  9349 

3711  9286 

3872  9220 

4033  9151 

4192  9079 

13 

48 

3551  9348 

3714  9285 

3875  9219 

4035  9150 

4195  9078 

12 

49 

3554  9347 

3716  9284 

3878  9218 

4038  9148 

4197  9077 

11 

50 

3557  9346 

3719  9283 

3881  9216 

4041  9147 

4200  9075 

1O 

51 

3559  9345 

3722  9282 

3883  9215 

4043.  9146 

4202  9074 

9 

52 

3562  9344 

3724  9281 

3886  9214 

4046  9145 

4205  9073 

8 

53 

3565  9343 

3727  9279 

3889  9213 

4049  9144 

4208  9072 

7 

54 

3567  9342 

3730  9278 

3891  9212 

4051  9143 

4210  9070 

6 

55 

3570  9341 

3733  9277 

3894  9211 

4054  9141 

4213  .9069 

5 

56 

3573  9340 

3735  9276 

3897  9210 

4057  9140 

4216  9068 

4 

57 

3576  9339 

3738  9275 

3899  9208 

4059  9139 

4218  9067 

3 

58 

3578  9338 

3741  9274 

3902  9207 

4062  9138 

4221  9066 

2 

59 

3581  9337 

3743  9273 

3905  9206 

4065  9137 

4224  9064 

1 

6O 

3584  9336 

3746  9272 

3907  9205 

4067  9135 

4226  9063 

0 

cos   sin 

cos  sin 

cos  sin 

cos  sin 

cos  sin 

f 

69° 

68° 

67° 

66° 

65° 

f 

NATURAL   SINES    AND    COSINES. 


57 


/ 

25° 

26° 

27° 

28° 

29° 

t 

sin   cos 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

o 

4226  9063 

4384  8988 

4540  8910 

4695  8829 

4848  8746 

6O 

1 

4229  9062 

4386  8987 

4542  8909 

4697  8828 

4851  8745 

59 

2 

4231  9061 

4389  8985 

4545  8907 

4700  8827 

4853  8743 

58 

3 

4234  9059 

4392  8984 

4548  8906 

4702  8825 

4856  8742 

57 

4 

.  .4237  9058 

4394  8983 

4550  8905 

4705  8824 

4858  8741 

56 

5 

4239  9057 

4397  8982 

4553  8903 

4708  8823 

4861  8739 

55 

6 

4242  9056 

4399  8980 

4555  8902 

4710  8821 

4863  8738 

54 

7 

4245  9054 

4402  8979 

4558  8901 

4713  8820 

4866  8736 

53 

8 

4247  9053 

4405  8978 

4561  8899 

4715  8819 

4868  8735 

52 

9 

4250  9052 

4407  8976 

4563  8898 

4718  8817 

4871  8733 

51 

1O 

4253  9051 

4410  8975 

4566  8897 

4720  8816 

4874  8732 

50 

11 

4255  9050 

4412  8974 

4568  8895 

4723  8814 

4876  8731 

49 

12 

4258  9048 

4415  8973 

4571  8894 

4726  8813 

4879  8729 

48 

13 

4260  9047 

4418  8971 

4574  8893 

4728  8812 

4881  8728 

47 

14 

4263  9046 

4420  8970 

4576  8892 

4731  8810 

4884  8726 

46 

15 

4266  9045 

4423  8969 

4579  8890 

4733  8809 

4886  8725 

45 

16 

4268  9043 

4425  8967 

4581  8889 

4736  8808 

4889  8724 

44 

17 

4271  9042 

4428  8966 

4584  8888 

4738  8806 

4891  8722 

43 

18 

4274  9041 

4431  8965 

4586  8886 

4741  8805 

4894  8721 

42 

19 

4276  9040 

4433  '  8964 

4589  8885 

4743  8803 

4896  8719 

41 

2O 

4279  9038 

4436  8962 

4592  8884 

4746  8802 

4899  8718 

40 

21 

4281  9037 

4439  8961 

4594  8882 

4749  8801 

4901  8716 

39 

22 

4284  9036 

4441  8960 

4597  8881 

4751  8799 

4904  8715 

38 

23 

4287  9035 

4444  8958 

4599  8879 

4754  8798 

4907  8714 

37 

24 

4289  9033 

4446  8957 

4602  8878 

4756  8796 

4909  8712 

36 

25 

4292  9032 

4449  8956 

4605  8877 

4759  8795 

4912  8711 

35 

26 

4295  9031 

4452  8955 

4607  8875 

4761  8794 

4914  8709 

34 

27 

4297  9030 

4454  8953 

4610  8874 

4764  8792 

4917  8708 

33 

28 

4300  9028 

4457  8952 

4612  8873 

4766  8791 

4919  8706 

32 

29 

4302  9027 

4459  8951 

4615  8871 

4759  8790 

4922  8705 

31 

30 

4305  9026 

4462  8949 

4617  8870 

4772  8788 

4924  8704 

3O 

31 

4308  9025 

4465  8948 

4620  8869 

4774  8787 

4927  8702 

29 

32 

4310  9023 

4467  8947 

4623  8867 

4777  8785 

4929  8701 

28 

33 

4313  9022 

4470  8945 

4625  8866 

4779  8784 

4932  8699 

27 

34 

4316  9021 

4472  8944 

4628  8865 

4782  8783 

4934  8698 

26 

35 

4318  9020 

4475  8943 

4630  8863 

4784  8781 

4937  8696 

25 

36 

4321  9018 

4478  8942 

4633  8862 

4787  8780 

4939  8695 

24 

37 

4323  9017 

4480  8940 

4636  8861 

4789  8778 

4942  8694 

23 

38 

4326  9016 

4483  8939 

4638  8859 

4792  8777 

4944  8692 

22 

39 

4329  9015 

4485  8938 

4641  8858 

4795  8776 

4947  8691 

21 

4O 

4331  9013 

4488  8936 

4643  8857 

4797  8774 

4950  8689 

20 

41 

4334  9012 

4491  8935 

4646  8855 

4800  8773 

4952  8688 

19 

42 

4337  9011 

4493  8934 

4648  8854 

4802  8771 

4955  8686 

18 

43 

4339  9010 

4496  8932 

4651  8853 

4805  8770 

4957  8685 

17 

44 

4342  9008 

4498  8931 

4654  8851 

•4807  8769 

4960  8683 

16 

45 

4344  9007 

4501  8930 

4656  8850 

4810  8767 

4962  8682 

15 

46 

4347  9006 

4504  8928 

4659  8849 

4812  8766 

4965  8681 

14 

47 

4350  9004 

4506  8927 

4661  8847 

4815  8764 

4967  8679 

13 

48 

4352  9003 

4509  8926 

4664  8846 

4818  8763 

4970  8678 

12 

49 

4355  9002 

4511  8925 

4666  8844 

4820  8762 

4972  8676 

11 

50 

4358  9001 

4514  8923 

4669  8843 

4823  8760 

4975  8675 

1O 

51 

4360  8999 

4517  8922 

4672  8842 

4825  8759 

4977  8673 

•9 

52 

4363  8998 

4519  8921 

4674  8840 

4828  8757 

4980  8672 

8 

53 

4365  8997 

4522  8919 

4677  8839 

4830  8756 

4982  8670 

7 

54 

4368  8996 

4524  8918 

4679  8838 

4833  8755 

4985  8669 

6 

55 

4371  8994 

4527  8917 

4682  8836 

4835  8753 

4987  8668 

5 

56 

4373  8993 

4530  8915 

4684  8835 

4838  8752 

4990  8666 

4 

57 

4376  8992 

4532  8914 

4687  8834 

4840  8750 

4992  8665 

3 

58 

4378  8990 

4535  8913 

4690  8832 

4843  8749 

4995  8663 

2 

59 

4381  8989 

4537  8911 

4692  8831 

4846  8748 

4997  8662 

1 

60 

4384  8988 

4540  8910 

4695  8829 

4848  8746 

5000  8660 

O 

cos   sin 

cos  sin 

cos  sin 

cos  sin 

cos   sin 

f 

64° 

63° 

62° 

61° 

60° 

f 

58 


NATURAL   SINES   AND   COSINES. 


/ 

3O° 

31° 

32° 

33° 

34° 

f 

sin   cos 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

0 

5000  8660 

5150  8572 

5299  8480 

5446  8387 

5592  8290 

60 

1 

5003  8659 

5153  8570 

5302  8479 

5449  8385 

5594  8289 

59 

2 

5005  8657 

5155  8569 

5304  8477 

5451  8384 

5597  8287 

58 

3 

5008  8656 

5158  8567 

5307  8476 

5454  8382 

5599  8285 

57 

4 

5010  8654 

5160  8566 

5309  8474 

5456  8380 

5602  8284 

56 

5 

5013  8653 

5163  8564 

5312  8473 

5459  8379 

5604  8282  " 

55 

6 

5015  8652 

5165  8563 

5314  8471 

5461  8377 

5606  8281 

54 

7 

5018  8650 

5168  8561 

5316  8470 

5463  8376 

5609  8279 

53 

8 

5020  8649 

5170  8560 

5319  8468 

5466  8374 

5611  8277 

52 

9 

5023  8647 

5173  8558 

5321  8467 

5468  8372 

5614  8276 

51 

1O 

5025  8646 

5175  8557 

5324  8465 

5471  8371 

5616  8274 

5O 

11 

5028  8644 

5178  8555 

5326  8463 

5473  8369 

5618  8272 

49 

12 

5030  8643 

5180  8554 

5329  8462 

5476  8368 

5621  8271 

48 

13 

5033  8641 

5183  8552 

5331  8460 

5478  8366 

5623  8269 

47 

14 

5035  8640 

5185  8551 

5334  8459 

5480  8364 

5626  8268 

46 

15 

5038  8638 

5188  8549 

5336  8457 

5483  8363 

5628  8266 

45 

16 

5040  8637 

5190  8548 

5339  8456 

5485  8361 

5630  8264 

44 

17 

5043  8635 

5193  8546 

5341  8454 

5488  8360 

5633  8263 

43 

18 

5045  8634 

5195  8545 

5344  8453 

5490  8358 

5635  8261 

42 

19 

5048  8632 

5198  8543 

5346  8451 

5493  8356 

5638  8259 

41 

20 

5050  8631 

5200  8542 

5348  8450 

5495  8355 

5640  8258 

4O 

21 

5053  8630 

5203  8540 

5351  8448 

5498  8353 

5642  8256 

39 

22 

5055  8628 

5205  8539 

5353  8446 

5500  8352 

5645  8254 

38 

23 

5058  8627 

5208  8537 

5356  8445 

5502  8350 

5647  8253 

37 

24 

5060  8625 

5210  8536 

5358  8443 

5505  8348 

5650  8251 

36 

25 

5063  8624 

5213  8534 

5361  8442 

5507  8347 

5652  8249 

35 

26 

5065  8622 

5215  8532 

5363  8440 

5510  8345 

5654  8248 

34 

27 

5068  8621 

5218  8531 

5366  8439 

5512  8344 

5657  8246 

33 

28 

5070  8619 

5220  8529 

5368  8437 

5515  8342 

5659  8245 

32 

29 

5073  8618 

5223  8528 

5371  8435 

5517  8340 

5662  8243 

31 

30 

5075  8616 

5225  8526 

5373  8434 

5519  8339 

5664  8241 

3O 

31 

5078  8615 

5227  8525 

5375  8432 

5522  8337 

5666  8240 

29 

32 

5080  8613 

5230  8523 

5378  8431 

5524  8336 

5669  8238 

28 

33 

5083  8612 

5232  8522 

5380  8429 

5527  8334 

5671  8236 

27 

34 

5085  8610 

5235  8520 

5383  8428 

5529  8332 

5674  8235 

26 

35 

5088  8609 

5237  8519 

5385  8426 

5531  8331 

5676  8233 

25 

36 

5090  8607 

5240  8517 

5388  8425 

5534  8329 

5678"  8231 

24 

37 

5093  8606 

5242  8516 

5390  8423 

5536  8328 

5681  8230 

23 

38 

5095  8604 

5245  8514 

5393  8421 

5539  8326 

5683  8228 

22 

39 

5098  8603 

5247  8513 

5395  8420 

5541  8324 

5686  8226 

21 

40 

5100  8601 

5250  8511 

5398  8418 

5544  8323 

5688  8225 

2O 

41 

5103  8600 

5252  8510 

5400  8417 

5546  8321 

5690  8223 

19 

42 

5105  8599 

5255  8508 

5402  8415 

5548  8320 

5693  8221 

18 

43 

5108  8597 

5257  8507 

5405  8414 

5551  8318 

5695  8220 

17 

44 

5110  8596 

5260  8505 

5407  8412 

5553  8316 

5698  8218 

16 

45 

5113  8594 

5262  8504 

5410  8410 

5556  8315 

5700  8216 

15 

46 

5115  8593 

5265  8502 

5412  8409 

5558  8313 

5702  8215 

14 

47 

5118  8591 

5267  8500 

5415  8407 

5561  8311 

5705  8213 

13 

48 

5120  8590 

5270  8499 

5417  8406 

5563  8310 

5707  8211 

12 

49 

5123  8588 

5272  8497 

5420  8404 

5565  8308 

5710  8210 

11 

50 

5125  8587 

5275  8496 

5422  8403 

5568  8307 

5712  8208 

10 

51 

5128  8585 

5277  8494 

5424  8401 

5570  8305 

5714  8207 

9 

52 
53 

5130  8584 
5133  8582 

5279  8493 
5282  8491 

5427  8399 
5429  8398 

5573  8303 
5575  8302 

5717  8205 
5719  8203 

8 

7 

54 

5135  8581 

5284  8490 

5432  8396 

5577  8300 

5721  8202 

6 

55 

5138  8579 

5287  8488 

5434  8395 

5580  8299 

5724  8200 

5 

56 

5140  8578 

5289  8487 

5437  8393 

5582  8297 

5726  8198 

4 

57 

5143  8576 

5292  8485 

5439  8391 

5585  8295 

5729  8197 

3 

58 

5145  8575 

5294  8484 

5442  8390 

5587  8294 

5731  8195 

2 

59 

S148  8573 

5297  8482 

5444  8388 

5590  8292 

5733  8193 

1 

60 

5150  8572 

5299  8480 

5446  8387 

5592  8290 

5736  8192 

O 

cos  sin 

cos  sin 

cos  sin 

cos  sin 

cos   sin 

f 

59° 

58° 

57° 

56° 

55° 

t 

NATURAL  SINES   AND   COSINES. 


59 


t 

35° 

30° 

37° 

38° 

39° 

t 

sin   cos 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

o 

5736  8192 

5878  8090 

6018  7986 

6157  7880 

6293  7771 

60 

1 

5738  8190 

5880  8088 

6020  7985 

6159  7878 

6295  7770 

59 

2 

5741  8188 

5883  8087 

'6023  7983 

6161  7877 

6298  7768 

58 

3 

5743  8187 

5885  8085 

6025  7981 

6163  7875 

6300  7766 

57 

4 

5745  8185 

5887  8083 

6027  7979 

6166  7873 

6302  7764 

56 

5 

5748  8183 

5890  8082 

6030  7978 

6168  7871 

6305  7762 

55 

6 

5750  8181 

5892  8080 

6032  7976 

6170  7869 

6307  7760 

54 

7 

5752  8180 

5894  8078 

6034  7974 

6173  7868 

6309  7759 

53 

8 

5755  8178 

5897  8076 

6037  7972 

6175  7866 

6311  7757 

52 

9 

5757  8176 

5899  8075 

6039  7971 

6177  7864 

6314  7755 

51 

10 

5760  8175 

5901  8073 

6041  7969 

6180  7862 

6316  7753 

50 

11 

5762  8173 

5904  8071 

6044  7967 

6182  7860 

6318  7751 

49 

12 

5764  8171 

5906  8070 

6046  7965 

6184  7859 

6320  7749 

48 

13 

5767  8170 

5908  8068 

6048  7964 

6186  7857 

6323  7748 

47 

14 

5769  8168 

5911  8066 

6051  7962 

6189  7855 

6325  7746 

46 

15 

5771  8166 

5913  8064 

6053  7960 

6191  7853 

6327  7744 

45 

16 

5774  8165 

5915  8063 

6055  7958 

6193  7851 

6329  7742 

44 

17 

5776  8163 

5918  8061 

6058  7956 

6196  7850 

6332  7740 

43 

18 

5779  8161 

5920  8059 

6060  7955 

6198  7848 

6334  7738 

42 

19 

5781  8160 

5922  8058 

6062  7953 

6200  7346 

6336  7737 

41 

2O 

5783  8158 

5925  8056 

6065  7951 

6202  7844 

6338  7735 

40 

21 

5786  8156 

5927  8054 

6067  7950 

6205  7842 

6341  7733 

39 

22 

5788  8155 

5930  8052 

6069  7948 

6207  7841 

6343  7731 

38 

23 

5790  8153 

5932  8051 

6071  7946 

6209  7839 

6345  7729 

37 

24 

5793  8151 

5934  8049 

6074  7944 

6211  7837 

6347  7727 

36 

25 

5795  8150 

5937  8047 

6076  7942 

6214  7835 

6350  7725 

35 

26 

5798  8148 

5939  8045 

6078  7941 

6216  7833 

6352  7724 

34 

27 

5800  8146 

5941  8044 

6081  7939 

6218  7832 

6354  7722 

33 

28 

5802  8145 

5944  8042 

6083  7937 

6221  7830 

6356  7720 

32 

29 

5805  8143 

5946  8040 

6085  7935 

6223  7828 

6359  7718 

31 

30 

5807  8141 

5948  8039 

6088  7934 

6225  7826 

6361  7716 

30 

31 

5809  8139 

5951  8037 

6090  7932 

6227  7824 

6363  7714 

29 

.32 

5812  8138 

5953  8035 

6092  7930 

6230  7822 

6365  7713 

28 

33 

5814  8136 

5955  8033 

6095  7928 

6232  7821 

6368  7711 

27 

34 

5816  8134 

5958  8032 

6097  7926 

6234  7819 

6370  7709 

26 

35 

5819  8133 

5960  8030 

6099  7925 

6237  7817 

6372  7707 

25 

36 

5821  8131 

5962  8028 

6101  7923 

6239  7815 

6374  7705 

24 

37 

5824  8129 

5965  8026 

6104  7921 

6241  7813 

6376  7703 

23 

38 

5826  8128 

5967  8025 

6106  7919 

6243  7812 

6379  7701 

22 

39 

5828  8126 

5969  8023 

6108  7918 

6246  7810 

6381  7700 

21 

4O 

5831  8124 

5972  8021 

61]  1  7916 

6248  7808 

6383  7698 

2O 

41 

5833  8123 

5974  8020 

6113  7914 

6250  7806 

6385  7696 

19 

42 

5835  8121 

5976  8018 

6115  7912 

6252  7804 

6388  7694 

18 

43 

5838  8119 

5979  8016 

6118  7910 

6255  7802 

6390  7692 

17 

44 

5840  8117 

5981  8014 

6120  7909 

6257  7801 

6392  7690 

16 

45 

5842  8116 

5983  8013 

6122  7907 

6259  7799 

6394  7688 

15 

46 

5845  8114 

5986  8011 

6124  7905 

6262  7797 

6397  7687 

14 

47 

5847  8112 

5988  8009 

6127  7903 

6264  7795 

6399  7685 

13 

48 

5850  8111 

5990  8007 

6129  7902 

6266  7793 

6401  7683 

12 

•  49 

5852  8109 

5993  8006 

6131  7900 

6268  7792 

6403  7681 

11 

50 

5854  8107 

5995  8004 

6134  7898 

6271  7790 

6406  7679 

10 

51 

5857  8106 

5997  8002 

6136  7896 

6273  7788 

6408  7677 

9 

52 

5859  8104 

6000  8000 

6138  7894 

6275  7786 

6410  7675 

8 

53 

5861  8102 

6002  7999 

6141  7893 

6277  7784 

6412  7674 

7 

54 

5864  8100 

6004  7997 

6143  7891 

6280  7782 

6414  7672 

6 

55 

5866  8099 

6007  7995 

6145  7889 

6282  7781 

6417  7670 

5 

56 

5868  8097 

6009  7993 

6147  7887 

6284  7779 

6419  7668 

4 

57 

5871  8095 

6011  7992 

6150  7885 

6286  7777 

6421  7666 

3 

58 

5873  8094 

6014  7990 

6152  7884 

6289  7775 

6423  7664 

2 

59 

5875  8092 

6016  7988 

6154  7882 

6291  7773 

6426  7662 

1 

6O 

5878  8090 

6018  7986 

6157  7880 

6293  7771 

6428  7660 

O 

cos  sin 

cos  sin 

cos  sin 

cos  sin 

cos   sin 

t 

54° 

53° 

52° 

51° 

50° 

t 

60 


NATURAL   SIXES   AND   COSINES. 


/ 

4O° 

41° 

42° 

43° 

44° 

f 

sin   cos 

sin  cos 

sin  cos 

sin  cos 

sin  cos 

o 

6428  7660 

6561  7547 

6691  7431 

6820  7314 

6947  7193 

60 

1 

6430  7659 

6563  7545 

6693  7430 

6822  7312 

6949  7191 

59 

2 

6432  7657 

6565  7543 

6696  7428 

6824  7310 

6951  7189 

58 

3 

6435  7655 

6567  7541 

6698  7426 

6826  7308 

6953  7187 

57 

4 

6437  7653 

6569  7539 

6700  7424 

6828  7306 

6955  7185 

56 

5 

6439  7651 

6572  7538 

6702  7422 

6831  7304 

6957  7183 

55 

6 

6441  7649 

6574  7536 

6704  7420 

6833  7302 

6959  7181 

54 

7 

6443  7647 

6576  7534 

6706  7418 

6835  7300 

6961  7179 

53 

8 

6446  7645 

6578  7532 

6709  7416 

6837  7298 

6963  7177 

52 

9 

6448  7644 

6580  7530 

6711  7414 

6839  7296 

6965  7175 

51 

1O 

6450  7642 

6583  7528 

6713  7412 

6841  7294 

6967  7173 

5O 

11 

6452  7640 

6585  7526 

6715  7410 

6843  7292 

6970  7171 

49 

12 

6455  7638 

6587  7524 

6717  7408 

6845  7290 

6972  7169 

48 

13 

6457  7636 

6589  7522 

6719  7406 

6848  7288 

6974  7167 

47 

14 

6459  7634 

6591  7520 

6722  7404 

6850  7286 

6976  7165 

46 

15 

6461  7632 

6593  7518 

6724  7402 

6852  7284 

6978  7163 

45 

16 

6463  7630 

6596  7516 

6726  7400 

6854  7282 

6980  7161 

44 

17 

6466  7629 

6598  7515 

6728  7398 

6856  7280 

6982  7159 

43 

18 

6468  7627 

6600  7513 

6730  7396 

6858  7278 

6984  7157 

42 

19 

6470  7625 

6602  7511 

6732  7394 

6860  7276 

6986  7155 

41 

20 

6472  7623 

6604  7509 

6734  7392 

6862  7274 

6988  7153 

40 

21 

6475  7621 

6607  7507  • 

6737  7390 

6865  7272 

6990  7151 

39 

22 

6477  7619 

6609  7505 

6739  7388 

6867  7270 

6992  7149 

38 

23 

6479  7617 

6611  7503 

6741  7387 

6869  7268 

6995  7147 

37 

24 

6481  7615 

6613  7501 

6743  7385 

6871  7266 

6997  7145 

36 

25 

6483  7613 

6615  7499 

6745  7383 

6873  7264 

6999  7143 

35 

26 

6486  7612 

6617  7497 

6747  7381 

6875  7262 

7001  7141 

34 

27 

6488  7610 

6620  7495 

6749  7379 

6877  7260 

7003  7139 

33 

28 

6490  7608 

6622  7493 

6752  7377 

6879  7258 

7005  7137 

32 

29 

6492  7606 

6624  7491 

6754  7375 

6881  7256 

7007  7135 

31 

30 

6494  7604 

6626  7490 

6756  7373 

6884  7254 

7009  7133 

30 

31 

6497  7602 

6628  7488 

6758  7371 

6886  7252 

7011  7130 

29 

32 

6499  7600 

6631  7486 

6760  7369 

6888  7250 

7013  7128 

28 

33 

6501  7598 

6633  7484 

6762  7367 

6890  7248 

7015  7126 

27 

34 

6503  7596 

6635  7482 

6764  7365 

6892  7246 

7017  7124 

26 

35 

6506  7595 

6637  7480 

6767  7363 

6894  7244' 

7019  7122 

25 

36 

6508  7593 

6639  7478 

6769  7361 

6896  7242 

7022  7120 

24 

37 

6510  7591 

6641  7476 

6771  7359 

6898  7240 

7024  7118 

23 

38 

6512  7589 

6644  7474 

6773  7357 

6900  7238 

7026  7116 

22 

39 

6514  7587 

6646  7472 

6775  7355 

6903  7236 

7028  7114 

21 

4O 

6517  758S 

6648  7470 

6777  7353 

6905  7234 

7030  7112 

2O 

41 

6519  7583 

6650  7468 

6779  7351 

6907  7232 

7032  7110 

19 

42 

6521  7581 

6652  7466 

6782  7349 

6909  7230 

7034  7108 

18 

43 

6523  7579 

6654  7464 

6784  7347 

6911  7228 

7036  7106 

17 

44 

6525  7578 

6657  7463 

6786  7345 

6913  7226 

7038  7104 

16 

45 

6528  7576 

6659  7461 

6788  7343 

6915  7224 

7040  7102 

15 

46 

6530  7574 

6661  7459 

6790  7341 

6917  7222 

7042  7100 

14 

47 

6532  7572 

6663  7457 

6792  7339 

6919  7220 

7044  7098 

13 

48 

6534  7570 

6665  7455 

6794  7337 

6921  7218 

7046  7096 

12 

49 

6536  7568 

6667  7453 

6797  7335 

6924  7216 

7048  7094 

11 

5O 

6539  7566 

6670  7451 

6799  7333 

6926  7214 

7050  7092 

10 

51 

6541  7564 

6672  7449 

6801  7331 

6928  7212 

7053  7090 

9 

52 

6543  7562 

6674  7447 

6803  7329 

6930  7210 

7055  7088 

8 

53 

6545  7560 

6676  7445 

6805  7327 

6932  7208 

7057  7085 

7 

54 

6547  7559 

6678  7443 

6807  7325 

6934  7206 

7059  7083 

6 

55 

6550  7557 

6680  7441 

6809  7323 

6936  7203 

7061  7081 

5 

56 

6552  7555 

6683  7439 

6811  7321 

6938  7201 

7063  7079 

4 

57 

6554  7553 

6685  7437 

6814  7319 

6940  7199 

7065  7077 

3 

58 

6556  7551 

6687  7435 

6816  7318 

6942  7197 

7067  7075 

2 

59 

6558  7549 

6689  7433 

6818  7316 

6944  7195 

7069  7073 

1 

60 

6561  7547 

6691  7431 

6820  7314 

6947  7193 

7071  7071 

O 

cos   sin 

cos  sin 

cos  sin 

cos  sin 

cos  sin 

t 

49° 

48° 

47° 

46° 

45° 

f 

NATURAL  TANGENTS  AND  COTANGENTS. 


61 


t 

0° 

1° 

2° 

3° 

4° 

t 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

0 

0000 

Infinite 

0175 

57.2900 

0349 

28.6363 

0524 

19.0811 

0699 

14.3007 

6O 

1 

0003 

3437.75 

0177 

56.3506 

0352 

28.3994 

0527 

18.9755 

0702 

14.2411 

59 

2 

0006 

1718.87 

0180 

55.4415 

0355 

28.1664 

0530 

188711 

0705 

14.1821 

58 

3 

0009 

1145.92 

0183 

54.5613 

0358 

27.9372 

0533 

18.7678 

0708 

14.1235 

57 

4 

0012 

859.436 

0186  53.7086 

0361 

27.7117 

0536 

18.6656 

0711 

14.0655 

56 

5 

0015 

687.549 

0189 

52.8821 

0364 

27.4899 

0539 

18.5645 

0714 

14.0079 

55 

6 

0017 

572.957 

0192 

52.0807 

0367 

27.2715 

0542 

18.4645 

0717 

13.9507 

54 

7 

0020 

491.106 

0195 

51.3032 

0370 

27.0566 

0544 

18.3655 

0720 

13.8940 

53 

8 

0023 

429.718 

0198 

50.5485 

0373 

26.8450 

0547 

18.2677 

0723 

13.8378 

52 

9 

0026 

381.971 

0201 

49.8157 

0375 

26.6367 

0550 

18.1708 

0726 

13.7821 

51 

1O 

0029 

343.774 

0204 

49.1039 

0378 

26.4316 

0553 

18.0750 

0729 

13.7267 

50 

11 

0032 

312.521 

0207 

48.4121 

0381 

26.2296 

0556 

17.9802 

0731 

13.6719 

49 

•  12 

0035 

286.478 

0209 

47.7395 

0384 

26.0307 

0559 

17.8863 

0734 

13.6174 

48 

13 

0038 

264.441 

0212 

47.0853 

0387 

25.8348 

0562 

17.7934 

0737 

13.5634 

47 

14 

0041 

245.552 

0215 

46.4489 

0390 

25.6418 

0565 

17.7015 

0740 

13.5098 

46 

15 

0044 

229.182 

0218 

45.8294 

0393 

25.4517 

0568 

17.6106 

0743 

13.4566 

45 

16 

0047 

214.858 

0221 

45.2261 

0396 

25.2644 

0571 

17.5205 

0746 

13.4039 

44 

17 

0049 

202.219 

0224 

44.6386 

0399 

25.0798 

0574 

17.4314 

0749 

13.3515 

43 

18 

0052 

190.984 

0227 

44.0661 

0402 

24.8978 

0577 

17.3432 

0752 

13.2996 

42 

19 

0055 

180.932 

0230 

43.5081 

0405 

24.7185 

0580 

17.2558 

0755 

13.2480 

41 

20 

0058 

171.885 

0233 

42.9641 

0407 

24.5418 

0582 

17.1693 

0758 

13.1969 

40 

21 

0061 

163.700 

0236 

42.4335 

0410 

24.3675 

0585 

17.0837 

0761 

13.1461 

39 

22 

0064 

156.259 

0239 

41.9158 

0413 

24.1957 

0588 

16.9990 

0764 

13.0958 

38 

23 

0067 

149.465 

0241 

41.4106 

0416 

24.0263 

0591 

16.9150 

0767 

13.0458 

37 

24 

0070 

143.237 

0244 

40.9174 

0419 

23.8593 

0594 

16.8319 

0769 

12.9962 

36 

25 

0073 

137.507 

0247 

40.4358 

0422 

23.6945 

0597 

16.7496 

0772 

12.9469 

35 

26 

0076 

132.219 

0250 

39.9655 

0425 

23.5321 

0600 

16.6681 

0775 

12.8981 

34 

27 

0079 

127.321 

0253 

39.5059 

0428 

23.3718 

0603 

16.5874 

0778 

12.8496 

33 

28 

0081 

122.774 

0256 

39.0568 

0431 

23.2137 

0606 

16.5075 

0781 

12.8014 

32 

29 

0084 

118.540 

0259 

38.6177 

0434 

23.0577 

0609 

16.4283 

0784 

12.7536 

31 

30 

0087 

114.589 

0262 

38.1885 

0437 

22.9038 

0612 

16.3499 

0787 

12.7062 

3O 

31 

0090 

110.892 

0265 

37.7686 

0440 

22.7519 

0615 

16.2722 

0790 

12.6591 

29 

32 

0093 

107.426 

0268 

37.3579 

0442 

22.6020 

0617 

16.1952 

0793 

12.6124 

28 

33 

0096 

104.171 

0271 

36.9560 

0445 

22.4541 

0620 

16.1190 

0796 

12.5660 

27 

34 

0099 

101.107 

0274 

36.5627 

0448 

22.3081 

0623 

16.0435 

0799 

12.5199 

26 

35 

0102 

98.2179 

0276 

36.1776 

0451 

22.1640 

0626 

15.9687 

0802 

12.4742 

25 

36 

0105 

95.4895 

0279 

358006 

0454 

22.0217 

0629 

15.8945 

0805 

12.4288 

24 

37 

0108 

92.9085 

0282 

35.4313 

0457 

21.8813 

0632 

15.8211 

0808 

12.3838 

23 

38 

0111 

90.4633 

0285 

35.0695 

0460 

21.7426 

0635 

15.7483 

0810 

12.3390 

22 

39 

0113 

88.1436 

0288 

34.7151 

0463 

21.6056 

0638 

15.6762 

0813 

12.2946 

21 

4O 

0116 

85.9398 

0291 

34.3678 

0466 

21.4704 

0641 

15.6048 

0816 

12.2505 

20 

41 

0119 

83.8435 

0294 

34.0273 

0469 

21.3369 

0644 

15.5340 

0819 

12.2067 

19 

42 

0122 

81.8470 

0297 

33.6935 

0472 

21.2049 

0647 

15.4638 

0822 

12.1632 

18 

43 

0125 

79.9434 

0300 

33.3662 

0475 

21.0747 

0650 

15.3943 

0825 

12.1201 

17 

44 

0128 

78.1263 

0303 

33.0452 

0477 

20.9460 

0653 

15.3254 

0828 

12.0772 

16 

45 

0131 

76.3900 

0306 

32.7303 

0480 

20.8188 

0655 

15.2571 

0831 

12.0346 

15 

46 

0134 

74.7292 

0308 

32.4213 

0483 

20.6932 

0658 

15.1893 

0834 

11.9923 

14 

47 

0137 

73.1390 

0311 

32.1181 

0486 

20.5691 

0661 

15.1222 

0837 

11.9504 

13 

48 

0140 

71.6151 

0314 

31.8205 

0489 

20.4465 

0664 

15.0557 

0840 

11.9087 

12 

49 

0143 

70.1533 

0317 

31.5284 

0492 

20.3253 

0667 

14.9898 

0843 

11.8673 

11 

5O 

0146 

68.7501 

0320 

31.2416 

0495 

20.2056 

0670 

14.9244 

0846 

11.8262 

1O 

51 

0148 

67.4019 

0323 

30.9599 

0498 

20.0872 

0673 

14.8596 

0849 

11.7853 

9 

52 

0151 

66.1055 

0326 

30.6833 

0501 

19.9702 

0676 

14.7954 

0851 

11.7448 

8 

53 

0154 

64.8580 

0329 

30.4116 

0504 

19.8546 

0679 

14.7317 

0854 

11.7045 

7 

54 

0157 

63.6567 

0332 

30.1446 

0507 

19.7403 

0682 

14.6685 

0857 

11.6645 

6 

55 

0160 

62.4992 

0335 

29.8823 

0509 

19.6273 

0685 

14.6059 

0860 

11.6248 

5 

56 

0163 

61.3829 

0338 

29.6245 

0512 

19.5156 

0688 

14.5438 

0863 

11.5853 

4 

57 

0166 

60.3058 

0340 

29.3711 

0515 

19.4051 

0690 

14.4823 

0866 

11.5461 

3 

58 

0169 

59.2659 

0343 

29.1220 

0518 

19.2959 

0693 

14.4212 

0869 

11.5072 

2 

59 

0172 

58.2612 

0346 

28.8771 

0521 

19.1879 

0696 

14.3607 

0872 

11.4685 

1 

60 

0175 

57.2900 

0349 

28.6363 

0524 

19.0811 

0699 

14.3007 

0875 

11.4301 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

f 

89° 

88° 

87° 

86° 

85° 

/ 

62 


NATURAL   TANGENTS    AND    COTANGENTS. 


t 

5° 

6° 

7° 

8° 

9° 

t 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

o 

0875  11.4301 

1051  9.5144 

1228  8.1443 

1405  7.1154 

1584  6.3138 

6O 

1 

0878  11.3919 

1054  9.4878 

1231  8.1248 

1408  7.1004 

1587  6.3019 

59 

2 

0881  11.3540 

1057  9.4614 

1234  8.1054 

1411  7.0855 

1590  6.2901 

58 

3 

0884  11.3163 

1060  9.4352 

1237  8.0860 

1414  7.0706 

1593  6.2783 

57 

4 

0887  11.2789 

1063  9.4090 

1240  8.0667 

1417  7.0558 

1596  6.2666 

56 

5 

0890  11.2417 

1066  9.3831 

1243  8.0476 

1420  7.0410 

1599  6.2549 

55 

6 

0892  11.2048 

1069  9.3572 

1246  8.0285 

1423  7.0264 

1602  6.2432 

54 

7 

0895  11.1681 

1072  9.3315 

1249  8.0095 

1426  7.0117 

1605  6.2316 

53 

8 

0898  11,1316 

1075  9.3060 

1251  7.9906 

1429  6.9972 

1608  6.2200 

52 

9 

0901  11.0954 

1078  9.2806 

1254  7.9718 

1432  6.9827 

1611  6.2085 

51 

10 

0904  11.0594 

1080  9.2553 

1257  7.9530 

1435  6.9682 

1614  6.1970 

50 

11 

0907  11.0237 

1083  9.2302 

1260  7.9344 

1438  6.9538 

1617  6.1856 

49 

12 

0910  10.9882 

1086  9.2052 

1263  7.9158 

1441  6.9395 

1620  6.1742 

48 

13 

0913  10.9529 

1089  9.1803 

1266  7.8973 

1444  6.9252 

1623  6.1628 

47 

14 

0916  10.9178 

1092  9.1555 

1269  7.8789 

1447  6.9110 

1626  6.1515 

46 

15 

0919  10.8829 

1095  9.1309 

1272  7.8606 

1450  6.8969 

1629  6.1402 

45 

16 

0922  10.8483 

1098  9.1065 

1275  7.8424 

1453  6.8828 

1632  6.1290 

44 

17 

0925  10.8139 

1101  9.0821 

1278  7.8243 

1456  6.8687 

1635  6.1178 

43 

18 

0928  10.7797 

1104  9.0579 

1281  7.8062 

1459  6.8548 

1638  6.1066 

42 

19 

0931  10.7457 

1107  9.0338 

1284  7.7883 

1462  6.8408 

1641  6.0955 

41 

2O 

0934  10.7119 

1110  9.0098 

1287  7.7704 

1465  6.8269 

1644  6.0844 

4O 

21 

0936  10.6783 

1113  8.9860 

1290  7.7525 

1468  6.8131 

1647  6.0734 

39 

22 

0939  10.6450 

1116  8.9623 

1293  7.7348 

1471  6.7994 

1650  6.0624 

38 

23 

0942  10.6118 

1119  8.9387 

1296  7.7171 

1474  6.7856 

1653  6.0514 

37 

24 

0945  10.5789 

1122  8.9152 

1299  7.6996 

1477  6.7720 

1655  6.0405 

36 

25 

0948  10.5462 

1125  88919 

1302  7.6821 

1480  6.7584 

1658  6.0296 

35 

26 

0951  10.5136 

1128  8.8686 

1305  7.6647 

1483  6.7448 

1661  6.0188 

34 

27 

0954  10.4813 

1131  88455 

1308  7.6473 

1486  6.7313 

1664  6.0080 

33 

28 

0957  10.4491 

1134  8.8225 

1311  7.6301 

1489  6.7179 

1667  5.9972 

32 

29 

0960  10.4172 

1136  8.7996 

1314  7.6129 

1492  6.7045 

1670  5.9865 

31 

30 

0963  10.3854 

1139  8.7769 

1317  7.5958 

1495  6.6912 

1673  5.9758 

30 

31 

0966  10.3538 

1142  8.7542 

1319  7.5787 

1497  6.6779 

1676  5.9651 

29 

32 

0969  10.3224 

1145  8.7317 

1322  7.5618 

1500  6.6646 

1679  5.9545 

28 

33 

0972  10.2913 

1148  8.7093 

1325  7.5449 

1503  6.6514 

1682  5.9439 

27 

34 

0975  10.2602 

1151  8.6870 

1328  7.5281 

1506  6.6383 

1685  5.9333 

26 

35 

0978  10.2294 

1154  8.6648 

1331  7.5113 

1509  6.6252 

1688  5.9228 

25 

36 

0981  10.1988 

1157  8.6427 

1334.  7.4947 

1512  6.6122 

1691  5.9124 

24 

37 

0983  10.1683 

1160  8.6208 

1337  7.4781 

1515  6.5992 

1694  5.9019 

23 

38 

0986  10.1381 

1163  8.5989 

1340  7.4615 

1518  6.5863 

1697  5.8915 

22 

39 

0989  10.1080 

1166  8.5772 

1343  7.4451 

1521  6.5734 

1700  5.8811 

21 

40 

0992  10.0780 

1169  8.5555 

1346  7.4287 

1524  6.5606 

1703  5.8708 

2O 

41 

0995  10.0483 

1172  8.5340 

1349  7.4124 

1527  6.5478 

1706,  5.8605 

19 

42 

0998  10.0187 

1175  8.5126 

1352  7.3962 

1530  6.5350 

1709  5.8502 

18 

43 

1001  9.9893 

1178  8.4913 

1355  7.3800 

1533  6.5223 

1712  5.8400 

17 

44 

1004  9.9601 

1181  8.4701 

1358  7.3639 

1536  6.5097 

1715  5.8298 

16 

45 

1007  9.9310 

1184  8.4490 

1361  7.3479 

1539  6.4971 

1718  5.8197 

15 

46 

1010  9.9021 

1187  8.4280 

1364  7.3319 

1542  6.4846 

1721  5.8095 

14 

47 

1013  9.8734 

1189  8.4071 

1367  7.3160 

1545  6.4721 

1724  5.7994 

13 

48 

1016  9.8448 

1192  8.3863 

1370  7.3002 

1548  6.4596 

1727  5.7894 

12 

49 

1019  9.8164 

1195  8.3656 

1373  7.2844 

1551  6.4472 

1730  5.7794 

11 

5O 

1022  9.7882 

1198  8.3450 

1376  7.2687 

1554  6.4348 

1733  5.7694 

1O 

51 

1025  9.7601 

1201  8.3245 

1379  7.2531 

1557  6.4225 

1736  5.7594 

9 

52 

1028  9.7322 

1204  8.3041 

1382  7.2375 

1560  6.4103 

1739  5.7495 

8 

53 

1030  9.7044 

1207  8.2838 

1385  7.2220 

1563  6.3980 

1742  5.7396 

7 

54 

1033  9.6768 

1210  8.2636 

1388  7.2066 

1566  6.3859 

1745  5.7297 

6 

55 

1036  9.6499 

1213  8.2434 

1391  7.1912 

1569  6.3737 

1748  5.7199 

5 

56 

1039  9.6220 

1216  8.2234 

1394  7.1759 

1572  6.3617 

1751  5.7101 

4 

57 

1042  9.5949 

1219  8.2035 

1397  7.1607 

1575  6.3496 

1754  5.7004 

3 

58 

1045  9.5679 

1222  8.1837 

1399  7.1455 

1578  6.3376 

1757  5.6906 

2 

59 

1048  9.5411 

1225  8.1640 

1402  7.1304 

1581  6.3257 

1760  5.6809 

1 

60 

1051  9.5144 

1228  8.1443 

1405  7.1154 

1584  6.3138 

1763  5.6713 

0 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

f 

84° 

83° 

82° 

81° 

8O° 

t 

NATURAL   TANGENTS   AND    COTANGENTS. 


63 


1 

1O° 

11° 

12° 

13° 

14° 

t 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

o 

1763  5.6713 

1944  5.1446 

2126  4.7046 

2309  4.3315 

2493  4.0108 

60 

1 

1766  5.6617 

1947  5.1366 

2129  4.6979 

2312  4.3257 

2496  4.0058 

59 

2 

1769  5.6521 

1950  5.1286 

2132  4.6912 

2315  4.3200 

2499  4.0009 

58 

3 

1772  5.6425 

1953  5.1207 

2135  4.6845 

2318  4.3143 

2503  3.9959 

57 

4 

1775  5.6330 

1956  5.1128 

2138  4.6779 

2321  4.3086 

2506  3.9910 

56 

5 

1778  5.6234 

1959  5.1049 

2141  4.6712 

2324  4.3029 

2509  3.9861 

55 

6 

1781  5.6140 

1962  5.0970 

2144  4.6646 

2327  4.2972 

2512  3.9812 

54 

7 

1784  5.6045 

1965  5.0892 

2147  4.6580 

2330  4.2916 

2515  3.9763 

53 

8 

1787  5.5951 

1968  5.0814 

2150  4.6514 

2333  4.2859 

2518  3.9714 

52 

9 

1790  5.5857 

1971  5.0736 

2153  4.6448 

2336  4.2803 

2521  3.9665 

51 

10 

1793  5.5764 

1974  5.0658 

2156  4.6382 

2339  4.2747 

2524  3.9617 

5O 

11 

1796  5.5671 

1977  5.0581 

2159  4.6317 

2342  4.2691 

2527  3.9568 

49 

12 

1799  5.5578 

1980  5.0504 

2162  4.6252 

2345  4.2635 

2530  3.9520 

48 

13 

1802  5.5485 

1983  5.0427 

2165  4.6187 

2349  4.2580 

2533  3.9471 

47 

14 

1805  5.5393 

1986  5.0350 

2168  4.6122 

2352  4.2524 

2537  3.9423 

46 

15 

1808  5.5301 

1989  5.0273 

2171  4.6057 

2355  4.2468 

2540  3.9375 

45 

16 

1811  5.5209 

1992  5.0197 

2174  4.5993 

2358  4.2413 

2543  3.9327 

44 

17 

1814  5.5118 

1995  5.0121 

2177  4.5928 

2361  4.2358 

2546  3.9279 

43 

18 

1817  5.5026 

1998  5.0045 

2180  4.5864 

2364  4.2303 

2549  3.9232 

42 

19 

1820  5.4936 

2001  4.9969 

2183  4.5800 

2367  4.2248 

2552  3.9184 

41 

2O 

1823  5.4845 

2004  4.9894 

2186  4.5736 

2370  4.2193 

2555  3.9136 

4O 

21 

1826  5.4755 

2007  4.9819 

2189  4.5673 

2373  4.2139 

2558  3.9089 

39 

22 

1829  5.4665 

2010  4.9744 

2193  4.5609 

2376  4.2084 

2561  3.9042 

38 

23 

1832  5.4575 

2013  4.9669 

2196  4.5546 

2379  4.2030 

2564  3.8995 

37 

24 

1835  5.4486 

2016  4.9594 

2199  4.5483 

2382  4.1976 

2568  3.8947 

36 

25 

1838  5.4397 

2019  4.9520 

2202  4.5420 

2385  4.1922 

2571  3.8900 

35 

26 

1841  5.4308 

2022  4.9446 

2205  4.5357 

2388  4.1868 

2574  3.8854 

34 

27 

1844  5.4219 

2025  4.9372 

2208  4.5294 

2392  4.1814 

2577  3.8807 

33 

28 

1847  5.4131 

2028  4.9298 

2211  4.5232 

2395  4.1760 

2580  3.8760 

32 

29 

1850  5.4043 

2031  4.9225 

2214  4.5169 

2398  4.1706 

2583  3.8714 

31 

30 

1853  5.3955 

2035  4.9152 

2217  4.5107 

2401  4.1653 

2586  3.8667 

30 

31 

1856  5.3868 

2038  4.9078 

2220  4.5045 

2404  4.1600 

2589  3.8621 

29 

32 

1859  5.3781 

2941  4.9006 

2223  4.4983 

2407  4.1547 

2592  3.8575 

28 

33 

1862  5.3694 

2044  4.8933 

2226  4.4922 

2410  4.1493 

2595  3.8528 

27 

34 

1865  5.3607 

2047  4.8860 

2229  4.4860 

2413  4.1441 

2599  3.8482 

26 

35 

1868  5.3521 

2050  4.8788 

2232  4.4799 

2416  4.1388 

2602  3.8436 

25 

36 

1871  5.3435 

2053  4.8716 

2235  4.4737 

2419  4.1335 

2605  3.8391 

24 

37 

1874  5.3349 

2056  4.8644 

2238  4.4676 

2422  4.1282 

2608  3.8345 

23 

38 

1877  5.3263 

2059  4.8573 

2241  4.4615 

2425  4.1230 

2611  3.8299 

22 

39 

1880  5.3178 

2062  4.8501 

2244  4.4555 

2428  4.1178 

2614  3.8254 

21 

40 

1883  5.3093 

2065  4.8430 

2247  4.4494 

2432  4.1126 

2617  3.8208 

20 

41 

1887  5.3008 

2068  4.8359 

2251  4.4434 

2435  4.1074 

2620  3.8163 

19 

42 

1890  5.2924 

2071  4.8288 

2254  4.4374 

2438  4.1022 

2623  3.8118 

18 

43 

1893  5.2839 

2074  4.8218 

2257  4.4313 

2441  4.0970 

2627  3.8073 

17 

44 

1896  5.2755 

2077  4.8147 

2260  4.4253 

2444  4.0918 

2630  3.8028 

16 

45 

1899  5.2672 

2080  4.8077 

2263  4.4194 

2447  4.0867 

2633  3.7983 

15 

46 

1902  5.2588 

2083  4.8007 

2266  4.4134 

2450  4.0815 

2636  3.7938 

14 

47 

1905  5.2505 

2086  4.7937 

2269  4.4075 

2453  4.0764 

2639  3.7893 

13 

48 

1908  5.2422 

2089  4.7867 

2272  4.4015 

2456  4.0713 

2642  3.7848 

12 

49 

1911  5.2339 

2092  4.7798 

2275  4.3956 

2459  4.0662 

2645  3.7804 

11 

50 

1914  5.2257 

2095  4.7729 

2278  4.3897 

2462  4.0611 

2648  3.7760 

10 

51 

1917  5.2174 

2098  4.7659 

2281  4.3838 

2465  4.0560 

2651  3.7715 

9 

52 

1920  5.2092 

2101  4.7591 

2284  4.3779 

2469  4.0509 

2655  3.7671 

8 

53 

1923  5.2011 

2104  4.7522 

2287  4.3721 

2472  4.0459 

2658  3.7627 

7 

54 

1926  5.1929 

2107  4.7453 

2290  4.3662 

2475  4.0408 

2661  3.7583 

6 

55 

1929  5.1848 

2110  4.7385 

2293  4.3604 

2478  4.0358 

2664  3.7539 

5 

56 

1932  5.1767 

2113  4.7317 

2296  4.3546 

2481  4.0308 

2667  3.7495 

4 

57 

1935  5.1686 

2116  4.7249 

2299  4.3488 

2484  4.0257 

2670  3.7451 

3 

58 

1938  5.1606 

2119  4.7181 

2303  4.3430 

2487  4.0207 

2673  3.7408 

2 

59 

1941  5.1526 

2123  4.7114 

2306  4.3372 

2490  4.0158 

2676  3.7364 

1 

60 

1944  5.1446 

2126  4.7046 

2309  4.3315 

2493  4.0108 

2679  3.7321 

O 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot  tan 

f 

79° 

78° 

77° 

76° 

75° 

i 

NATURAL  TANGENTS   AND   COTANGENTS. 


f 

15° 

16° 

17° 

18° 

19° 

r 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

o 

2679  3.7321 

2867  3.4874 

3057  3.2709 

3249  3.0777 

3443  2.9042 

6O 

1 

2683  3.7277 

2871  3.4836 

3060  3.2675 

3252  3.0746 

3447  2.9015 

59 

2 

2686  3.7234 

2874  3.4798 

3064  3.2641 

3256  3.0716 

3450  2.8987 

58 

3 

2689  3.7191 

2877  3.4760 

3067  3.2607 

3259  3.0686 

3453  2.8960 

57 

4 

2692  3.7148 

2880  3.4722 

3070  3.2573 

3262  3.0655 

3456  2.8933 

56 

5 

2695  3.7105 

2883  3.4684 

3073  3.2539 

3265  3.0625 

3460  2.8905 

55 

6 

2698  3.7062 

2886  3.4646 

3076  3.2506 

3269  3.0595 

3463  2.8878 

54 

7 

2701  3.7019 

2890  3.4608 

3080  3.2472 

3272  3.0565 

3466  2.8851 

53 

8 

2704  3.6975 

2893  3.4570 

3083  3.2438 

3275  3.0535 

3469  2.8824 

52 

9 

2708  3.6933 

2896  3.4533 

3086  3.2405 

3278  3.0505 

3473  2.8797 

51 

10 

2711  3.6891 

2899  3.4495 

3089  3.2371 

3281  3.0475 

3476  2.8770 

50 

11 

2714  3.6848 

2902  3.4458 

3092  3.2338 

3285  3.0445 

3479  2.8743 

49 

12 

2717  3.6806 

2905  3.4420 

3096  3.2305 

3288  3.0415 

3482  2.8716 

48 

13 

2720  3.6764 

2908  3.4383 

3099  3.2272 

3291  3.0385 

3486  2.8689 

47 

14 

2723  3.6722 

2912  3.4346 

3102  3.2238 

3294  3.0356 

3489  2.8662  . 

46 

15 

2726  3.6680 

2915  3.4308 

3105  3.2205 

3298  3.0326 

3492  2.8636 

45 

16 

2729  3.6638 

2918  3.4271 

3108  3.2172 

3301  3.0296 

3495  2.8609 

44 

17 

2733  3.6596 

2921  3.4234 

3111  3.2139 

3304  3.0267 

3499  2.8582 

43 

18 

2736  3.6554 

2924  3.4197 

3115  3.2106 

3307  3.0237 

3502  2.8556 

42 

19 

2739  3.6512 

2927  3.4160 

3118  3.2073 

3310  3.0208 

3505  2.8529 

41 

2O 

2742  3.6470 

2931  3.4124 

3121  3.2041 

3314  3.0178 

3508  2.8502 

4O 

21 

2745  3.6429 

2934  3.4087 

3124  3.2008 

3317  3.0149 

3512  2.8476 

39 

22 

2748  3.6387 

2937  3.4050 

3127  3.1975 

3320  3.0120 

3515  2.8449 

38 

23 

2751  3.6346 

2940  3.4014 

3131  3.1943 

3323  3.0090 

3518  2.8423 

37 

24 

2754  3.6305 

2943  3.3977 

3134  3.1910 

3327  3.0061 

3522  2.8397 

36 

25 

2758  3.6264 

2946  3.3941 

3137  3.1878 

3330  3.0032 

3525  2.8370 

35 

26 

2761  3.6222 

2949  3.3904 

3140  3.1845 

3333  3.0003 

3528  2.8344 

34 

27 

2764  3.6181 

2953  3.3868 

3143  3.1813 

3336  2.9974 

3531  2.8318 

33 

28 

2767  3.6140 

2956  3.3832 

3147  3.1780 

3339  2.9945 

3535  2.8291 

32 

29 

2770  3.6100 

2959  3.3796 

3150  3.1748 

3343  29916 

3538  2.8265 

31 

30 

2773  3.6059 

2962  3.3759 

3153  3.1716 

3346  2.9887 

3541  2.8239 

3O 

31 

2776  3.6018 

2965  3.3723 

3156  3.1684 

3349  2.9858 

3544  2.8213 

29 

32 

2780  3.5978 

2968  3.3687 

3159  3.1652 

3352  2.9829 

3548  2.8187 

28 

33 

2783  3.5937 

2972  3.3652 

3163  3.1620 

3356  2,9800 

3551  2.8161 

27 

34 

2786  3.5897 

2975  3.3616 

3166  3.1588 

3359  2.9772 

3554  2.8135 

26 

35 

2789  3.5856 

2978  3.3580 

3169  3.1556 

3362  2.9743 

3558  2.8109 

25 

36 

2792  3.5816 

2981  3.3544 

3172  3.1524 

3365  2.9714 

3561  2.8083 

24 

37 

2795  3.5776 

2984  3.3509 

3175  3.1492 

3369  2.9686 

3564  2.8057 

23 

38 

2798  3.5736 

2987  3.3473 

3179  3.1460 

3372  2.9657 

3567  2.8032 

22 

39 

2801  3.5696 

2991  3.3438 

3182  3.1429 

3375  2.9629 

3571  2.8006 

21 

40 

2805  3.5656 

2994  3.3402 

3185  3.1397 

3378  2.9600 

3574  2.7980 

2O 

41 

2808  3.5616 

2997  3.3367 

3188  3.1366 

3382  2.9572 

3577  2.7955 

19 

42 

2811  3.5576 

3000  3.3332 

3191  3.1334 

3385  2.9544 

3581  2.7929 

18 

43 

2814  3.5536 

3003  3.3297 

3195  3.1303 

3388  2.9515 

3584  2.7903 

17 

44 

2817  3.5497 

3006  3.3261 

3198  3.1271 

3391  2.9487 

3587  2.7878 

16 

45 

2820  3.5457 

3010  3.3226 

3201  3.1240 

3395  2.9459 

3590  2.7852 

15 

46 

2823  3.5418 

3013  3.3191 

3204  3.1209 

3398  2.9431 

3594  2.7827 

14 

47 

2827  3.5379 

3016  3.3156 

3207  3.1178 

3401  2.9403 

3597  2.7801 

13 

48 

2830  3.5339 

3019  3.3122 

3211  3.1146 

3404  2.9375 

3600  2.7776 

12 

49 

2833  3.5300 

3022  3.3087 

3214  3.1115 

3408  2.9347 

3604  2.7751 

11 

50 

2836  3.5261 

3026  3.3052 

3217  3.1084 

3411  2.9319 

3607  2.7725 

10 

51 

2839  3.5222 

3029  3.3017 

3220  3.1053 

3414  2.9291 

•  3610  2.7700 

9 

52 

2842  3.5183 

3032  3.2983 

3223  3.1022 

3417  2.9263 

3613  2.7675 

8 

53 

2845  3.5144 

3035  3.2948 

3227  3.0991 

3421  2.9235 

3617  2.7650 

7 

54 

2849  3.5105 

3038  3.2914 

3230  3.0961 

3424  2.9208 

3620  2.7625 

6 

55 

2852  3.5067 

3041  3.2880 

3233  3.0930 

3427  2.9180 

3623  2.7600 

5 

56 

2855  3.5028 

3045  3.2845 

3236  3.0899 

3430  2.9152 

3627  2.7575 

4 

57 

2858  3.4989 

3048  3.2811 

3240  3.0868 

3434  2.9125 

3630  2.7550 

3 

58 

2861  3.4951 

3051  3.2777 

3243  3.0838 

3437  2.9097 

3633  2.7525 

2 

59 

2864  3.4912 

3054  3.2743 

3246  3.0807 

3440  2.9070 

3636  2.7500 

1 

60 

2867  3.4874 

3057  3.2709 

3249  3.0777 

3443  2.9042 

3640  2.7475 

O 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

t 

74° 

73° 

72° 

71° 

70° 

9 

NATURAL  TANGENTS   AND   COTANGENTS. 


65 


t 

2O° 

21° 

22° 

23° 

24° 

t 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

o 

3640  2.7475 

3839  2.6051 

4040  2.4751 

4245  2.3559 

4452  2.2460 

6O 

1 

3643  2.7450 

3842  2.6028 

4044  2.4730 

4248  2.3539 

4456  2.2443 

59 

2 

3646  2.7425 

3845  2.6006 

4047  2.4709 

4252  2.3520 

4459  2.2425 

58 

3 

3650  2.7400 

3849  2.5983 

4050  2.4689 

4255  2.3501 

4463  2.2408 

57 

4 

3653  2.7376 

3852*  2.5961 

4054  2.4668 

4258  2.3483 

4466  2.2390 

56 

5 

3656  2.7351 

3855  2.5938 

4057  2.4648 

4262  2.3464 

4470  2.2373 

55 

6 

3659  2.7326 

3859  2.5916 

4061  2.4627 

4265  2.3445 

4473  2.2355 

54 

7 

3663  2.7302 

3862  2.5893 

4064  2.4606 

4269  2.3426 

4477  2.2338 

53 

8 

3666  2.7277 

3865  2.5871 

4067  2.4586 

4272  2.3407 

4480  2.2320 

52 

9 

3669  2.7253 

3869  2.5848 

4071  2.4566 

4276  2.3388 

4484  2.2303 

51 

1C 

3673  2.7228 

3872  2.5826 

4074  2.4545 

4279  2.3369 

4487  2.2286 

50 

11 

3676  2.7204 

3875  2.5864 

4078  2.4525 

4283  2.3351 

4491  2.2268 

49 

12 

3679  2.7179 

3879  2.5782 

4081  2.4504 

4286  2.3332 

4494  2.2251 

48 

13 

3683  2.7155 

3882  2.5759 

4084  2.4484 

4289  2.3313 

4498  2.2234 

47 

14 

3686  2.7130 

3885  2.5737 

4088  2.4464 

4293  2.3294 

4501  2.2216 

46 

15 

3689  2.7106 

3889  2.5715 

4091  2.4443 

4296  2.3276 

4505  2.2199 

45 

16 

3693  2.7082 

3892  2.5693 

4095  2.4423 

4300  2.3257 

4508  2.2182 

44 

17 

3696  2.7058 

3895  2.5671 

4098  2.4403 

4303  2.3238 

4512  2.2165 

43 

18 

3699  2.7034 

3899  2.5649 

4101  2.4383 

4307  2.3220 

4515  2.2148 

42 

19 

3702  2.7009 

3902  2.5627 

4105  2.4362 

4310  2.3201 

4519  2.2130 

41 

2O 

3706  2.6985 

3906  2.5605 

4108  2.4342 

4314  2.3183 

4522  2.2113 

40 

21 

3709  2.6961 

3909  2.5533 

4111  2.4322 

4317  2.3164 

4526  2.2096 

39 

22 

3712  2.6937 

3912  2.5561 

4115  2.4302 

4320  2.3146 

4529  2.2079 

38 

23 

3716  2.6913 

3916  2.5539 

4118  2.4282 

4324  2.3127 

4533  2.2062 

37 

24 

3719  2.6889 

3919  2.5517 

4122  2.4262 

4327  2.3109 

4536  2.2045 

36 

25 

3722  2.6865 

3922  2.5495 

4125  2.4242 

4331  2.3090 

4540  2.2028 

35 

26 

3726  2.6841 

3926  2.5473 

4129  2.4222 

4334  2.3072 

4543  2.2011 

34 

27 

3729  2.6818 

3929  2.5452 

4132  2.4202 

4338  2.3053 

4547  2.1994 

33 

28 

3732  2.6794 

3932  2.5430 

4135  2.4182 

4341  2.3035 

4550  2.1977 

32 

29 

3736  2.6770 

3936  2.5408 

4139  2.4162 

4345  2.3017 

4554  2.1960 

31 

3O 

3739  2.6746 

3939  2.5386 

4142  2.4142 

4348  2.2998 

4557  2.1943 

3O 

31 

3742  2.6723 

3942  2.5365 

4146  2.4122 

4352  2.2980 

4561  2.1926 

29 

32 

3745  2.6699 

3946  2.5343 

4149  2.4102 

4355  2.2962 

4564  2.1909 

28 

33 

3749  2.6675 

3949  2.5322 

4152  2.4083 

4359  2.2944 

4568  2.1892 

27 

34 

3752  .  2.6652 

3953  2.5300 

4156  2.4063 

4362  2.2925 

4571  2.1876 

26 

35 

3755  2.6628 

3956  2.5279 

4159  2.4043 

4365  2.2907 

4575  2.1859 

25 

36 

3759  2.6605 

3959  2.5257 

4163  2.4023 

4369  2.2889 

4578  2.1842 

24 

37 

3762  2.6581 

3963  2.5236 

4166  2.4004 

4372  2.2871 

4582  2.1825 

23 

38 

3765  2.6558 

3966  2.5214 

4169  2.3984 

4376  2.2853 

4585  2.1808 

22 

39 

3769  2.6534 

3969  2.5193 

4173  2.3964 

4379  2.2835 

4589  2.1792 

21 

40 

3772  2.6511 

3973  2.5172 

4176  2.3945 

4383  2.2817 

4592  2.1775 

2O 

41 

3775  2.6488 

3976  2.5150 

4180  2.3925 

4386  2.2799 

4596  2.1758 

19 

42 

3779  2.6464 

3979  2.5129 

4183  2.3906 

4390  2.2781 

4599  2.1742 

18 

43 

3782  2.6441 

3983  2.5108 

4187  2.3886 

4393  2.2763 

4603  2.1725 

17 

44 

3785  2.6418 

3986  2.5086 

4190  2.3867 

4397  2.2745 

4607  2.1708 

16 

45 

3789  2.6395 

3990  2.5065 

4193  2.3847 

4400  2.2727 

4610  2.1692 

15 

46 

3792  2.6371 

3993  2.5044 

4197  2.3828 

4404  2.2709 

4614  2.1675 

14 

47 

3795  2.6348 

3996  2.5023 

4200  2.3808 

4407  2.2691 

4617  2.1659 

13 

48 

3799  2.6325 

4000  2.5002 

4204  2.3789 

4411  2.2673 

4621  2.1642 

12 

49 

3802  2.6302 

4003  2.4981 

4207  2.3770 

4414  2.2655 

4624  2.1625 

11 

5O 

3805  2.6279 

4006  2.4960 

4210  2.3750 

4417  2.2637 

4628  2.1609 

1O 

51 

3809  2.6256 

4010  2.4939 

4214  2.3731 

4421  2.2620 

4631  2.1592 

9 

52 

3812  2.6233 

4013  2.4918 

4217  2.3712 

4424  2.2602 

4635  2.1576 

8 

53 

3815  2.6210 

4017  2.4897 

4221  2.3693 

4428  2.2584 

4638  2.1560 

7 

54 

3819  2.6187 

4020  2.4876 

4224  2.3673 

4431  2.2566 

4642  2.1543 

6 

55 

3822  2.6165 

4023  2.4855 

4228  2.3654 

4435  2.2549 

4645  2.1527 

5 

56 

3825  26142 

4027  2.4834 

4231  2.3635 

4438  2.2531 

4649  2.1510 

4 

57 

3829  2.6119 

4030  2.4813 

4234  2.3616 

4442  2.2513 

4652  2.1494 

3 

58 

3832  2.6096 

4033  2.4792 

4238  2.3597 

4445  2.2496 

4656  2.1478 

2 

59 

3835  2.6074 

4037  2.4772 

4241  2.3578 

4449  2.2478 

4660  2.1461 

1 

6O 

3839  2.6051 

4040  2.4751 

4245  2.3559 

4452  2.2460 

4663  2.1445 

0 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot  tan 

t 

69° 

68° 

67° 

66° 

65° 

t 

NATURAL  TANGENTS   AND   COTANGENTS. 


f 

25° 

26° 

27° 

28° 

29° 

T 

tan  cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

o 

46t>3  2.1445 

4877  2.0503 

5095  1.9626 

5317  1.8807 

5543  1.8040 

6O 

1 

4667  2.1429 

4881  2.0488 

5099  1.9612 

5321  1.8794 

5547  1.8028 

59 

2 

4670  2.1413 

4885  2.0473 

5103  1.9598 

5325  1.8781 

5551  1.8016 

58 

3 

4674  2.1396 

4888  2.0458 

5106  1.9584 

5328  1,8768 

5555  1.8003 

57 

4 

4677  2.1380 

4892  2.0443 

5110  1.9570 

5332  1.8755 

5558  1.7991 

56 

5 

4681  2.1364 

4895  2.0428 

5114  1.9556 

5336  1.8741 

5562  1.7979 

55 

6 

4684  2.1348 

4899  2.0413 

5117  1.9542 

5340  1.8728 

5566  1.7966 

54 

7 

4688  2.1332 

4903  2.0398 

5121  1.9528 

5343  1.8715 

5570  1.7954 

53 

8 

4691  2.1315 

4906  2.0383 

5125  1.9514 

5347  1.8702 

5574  1.7942 

52 

9 

4695  2.1299 

4910  2.0368 

5128  1.9500 

5351  1.8689 

5577  1.7930 

51 

1C 

4699  2.1283 

4913  2.0353 

5132  1.9486 

5354  1.8676 

5581  1.7917 

50 

11 

4702  2.1267 

4917  2.0338 

5136  1.9472 

5358  1.8663 

5585  1.7905 

49 

12 

4706  2.1251 

4921  2.0323 

5139  1.9458 

5362  1.8650 

5589  1.7893 

48 

13 

4709  2.1235 

4924  2.0308 

5143  1.9444 

5366  1.8637 

5593  1.7881 

47 

14 

4713  2.1219 

4928  2.0293 

5147  1.9430 

5369  1.8624 

5596  1.7868 

46 

15 

4716  2.1203 

4931  2.0278 

5150  1.9416 

5373  1.8611 

5600  1.7856 

45 

16 

4720  2.1187 

4935  2.0263 

5154  1.9402 

5377  1.8598 

5604  1.7844 

44 

17 

4723  2.1171 

4939  2.0248 

5158  1.9388 

5381  1.8585 

5608  1.7832 

43 

18 

4727  2.1155 

4942  2.0233 

5161  1.9375 

5384  1.8572 

5612  1.7820 

42 

19 

4731  2.1139 

4946  2.0219 

5165  1.9361 

5388  1.8559 

5616  1.7808 

41 

20 

4734  2.1123 

4950  2.0204 

5169  1.9347 

5392  1.8546 

5619  1.7796 

4O 

21 

4738  2.1107 

4953  2.0189 

5172  1.9333 

5396  1.8533 

5623  1.7783 

39 

22 

4741  2.1092 

4957  2.0174 

5176  1.9319 

5399  1.8520 

5627  1.7771 

38 

23 

4745  2.1076 

4960  2.0160 

5180  1.9306 

5403  1.8507 

5631  1.7759 

37 

24 

4748  2.1060 

4964  2.0145 

5184  1.9292 

5407  1.8495 

5635  1.7747 

36 

25 

4752  2.1044 

496S  2.0130 

5187  1.9278 

5411  1.8482 

5639  1.7735 

35 

26 

4755  2.1028 

4971  2.0115 

5191  1.9265 

5415  1.8469 

5642  1.7723 

34 

27 

4759  2.1013 

4975  2.0101 

5195  1.9251 

5418  1.8456 

5646  1.7711 

33 

28 

4763  2.0997 

4979  2.0086 

5198  1.9237 

5422  1.8443 

5650  1.7699 

32 

29 

4766  2.0981 

4982  2.0072 

5202  1.9223 

5426  1.8430 

5654  1.7687 

31 

30 

4770  2.0965 

4986  2.0057 

5206  1.9210 

5430  1.8418 

5658  1.7675 

3O 

31 

4773  2.0950 

4989  2.0042 

5209  1.9196 

5433  1.8405 

5662  1.7663 

29 

32 

4777  2.0934 

4993  2.0028 

5213  1.9183 

5437  1.8392 

5665  1.7651 

28 

33 

4780  2.0918 

4997  2.0013 

5217  1.9169 

5441  1.8379 

5669  1.7639 

27 

34 

4784  2.0903 

5000  1.9999 

5220  1.9155 

5445  1.8367 

5673  1.7627 

26 

35 

4788  2.0887 

5004  1.9984 

5224  1.9142 

5448  1.8354 

5677  1.7615 

25 

36 

4791  2.0872 

5008  1.9970 

5228  1.9128 

5452  1.8341 

5681  1.7603 

24 

37 

4795  2.0856 

5011  1.9955 

5232  1.9115 

5456  1.8329 

5685  1.7591 

23 

38 

4798  2.0840 

5015  1.9941 

5235  1.9101 

5460  1.8316 

5688  1.7579 

22 

39 

4802  2.0825 

5019  1.9926 

5239  1.9088 

5464  1.8303 

5692  1.7567 

21 

4O 

4806  2.0809 

5022  1.9912 

5243  1.9074 

5467  1.8291 

5696  1.7556 

20 

41 

4809  2.0794 

5026  1.9897 

5246  1.9061 

5471  1.8278 

5700  1.7544 

19 

42 

4813  2.0778 

5029  1.9883 

5250  1.9047 

5475  1.8265 

5704  1.7532 

18 

43 

4816  2.0763 

5033  1.9868 

5254  1.9034 

5479  1.8253 

5708  1.7520 

17 

44 

4820  2.0748 

5037  1.9854 

5258  1.9020 

5482  1.8240 

5712  1.7508 

16 

45 

4823  2.0732 

5040  1.9840 

5261  1.9007 

5486  1.8228 

5715  1.7496 

15 

46 

4827  2.0717 

5044  1.9825 

5265  1.8993 

5490  1.8215 

5719  1.7485 

14 

47 

4831  2.0701 

5048  1.9811 

5269  1.8980 

5494  1.8202 

5723  1.7473 

13 

48 

4834  2.0686 

5051  1.9797 

5272  1.8967 

5498  1.8190 

5727  1.7461 

12 

49 

4838  2.0671 

5055  1.9782 

5276  1.8953 

5501  1.8177 

5731  1.7449 

11 

5O 

4841  2.0655 

5059  1.9768 

5280  1.8940 

5505  1.8165 

5735  1.7437 

10 

51 

4845  2.0640 

5062  1.9754 

5284  1.8927 

5509  1.8152 

5739  1.7426 

9 

52 

4849  2.0625 

5066  1.9740 

5287  1.8913 

5513  1.8140 

5743  1.7414 

8 

53 

4852  2.0609 

5070  1.9725 

5291  1.8900 

5517  1.8127 

5746  1.7402 

7 

54 

4856  2.0594 

5073  1.9711 

5295  1.8887 

5520  1.8115 

5750  1.7391 

6 

55 

4859  2.0579 

5077  1.9697 

5298  1.8873 

5524  1.8103 

5754  1.7379 

5 

56 

4863  20564 

5081  1.9683 

5302  1.8860 

5528  1.8090 

5758  1.7367 

4 

57 

4867  2.0549 

5084  1.9669 

5306  1.8847 

5532  1.8078 

5762  1.7355 

3 

58 

4870  2.0533 

5088  1.9654 

5310  1.8834 

5535  1.8065 

5766  1.7344 

2 

59 

4874  2.0518 

5092  1.9640 

5313  1.8820 

5539  1.8053 

5770  1.7332 

1 

60 

,4877  2.0503 

5095  1.9626 

5317  1.8807 

5543  1.8040 

5774  1.7321 

O 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

f 

64° 

63° 

62° 

61° 

6O° 

t 

NATURAL  TANGENTS  AND  COTANGENTS. 


67 


t 

3O° 

31° 

32° 

33° 

34° 

t 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

o 

5774  1.7321 

6009  1.6643 

6249  1.6003 

6494  1.5399 

6745  1.4826 

6O 

1 

5777  1.7309 

6013  1.6632 

6253  1.5993 

6498  1.5389 

6749  1.4816 

59 

2 

5781  1.7297 

6017  1.6621 

6257  1.5983 

6502  1.5379 

6754  1.4807 

58 

3 

5785  1.7286 

6020  1.6610 

6261  1.5972 

6506  1.5369 

6758  1.4798 

57 

4 

5789  1.7274 

6024  1.6599 

6265  1.5962 

6511  1.5359 

6762  1.4788 

56 

5 

5793  1.7262 

6028  1.6588 

6269  1.5952 

6515  1.5350 

6766  1.4779 

55 

6 

5797  1.7251 

6032  1.6577 

6273  1.5941 

6519  1.5340 

6771  1.4770 

54 

7 

5801  1.7239 

6036  1.6566 

6277  1.5931 

6523  1.5330 

6775  1.4761 

53 

8 

5805  1.7228 

6040  1.6555 

6281  1.5921 

6527  1.5320 

6779  1.4751 

52 

9 

5808  1.7216 

6044  1.6545 

6285  1.5911 

6531  1.5311 

6783  1.4742 

51 

1O 

5812  1.7205 

604S  1.6534 

6289  1.5900 

6536  1.5301 

6787  1.4733 

5O 

11 

5816  1.7193 

6052  1.6523 

6293  1.5890 

6540  1.5291 

6792  1.4724 

49 

12 

5820  1.7182 

6056  1.6512 

6297  1.5880 

6544  1.5282 

6796  1.4715 

48 

13 

5824  1.7170 

6060  1.6501 

6301  1.5869 

6548  1.5272 

6800  1.4705 

47 

14 

5828  1.7159 

6064  1.6490 

6305  1.5859 

6552  1.5262 

6805  1.4696 

46 

15 

5832  1.7147 

6068  1.6479 

6310  1.5849 

6556  1.5253 

6809  1.4687 

45 

16 

5836  1.7136 

6072  1.6469 

6314  1.5839 

6560  1.5243 

6813  1.4678 

44 

17 

5840  1.7124 

6076  1.6458 

6318  1.5829 

6565  1.5233 

6817  1.4669 

43 

18 

5844  1.7113 

6080  1.6447 

6322  1.5818 

6569  1.5224 

6822  1.4659 

42 

19 

5847  1.7102 

6084  1.6436 

6326  1.5808 

6573  1.5214 

6826  1.4650 

41 

20 

5851  1.7090 

6088  1.6426 

6330  1.5798 

6577  1.5204 

6830  1.4641 

4O 

21 

5855  1.7079 

6092  1.6415 

6334  1.5788 

6581  1.5195 

6834  1.4632 

39 

22 

5859  1.7067 

6096  1.6404 

6338  1.5778 

6585  1.5185 

6839  1.4623 

38 

23 

5863  1.7056 

6100  1.6393 

6342  1.5768 

6590  1.5175 

6843  1.4614 

37 

24 

5867  1.7045 

6104  1.6383 

6346  1.5757 

6594  1.5166 

6847  1.4605 

36 

25 

5871  1.7033 

6108  1.6372 

6350  1.5747 

6598  .1.5156 

6851  1.4596 

35 

26 

5875  1.7022 

6112  1.6361 

6354  1.5737 

6602  1.5147 

6856  1.4586 

34 

27 

5879  1.7011 

6116  1.6351 

6358  1.5727 

6606  1.5137 

6860  1.4577 

33 

28 

5883  1.6999 

6120  1.6340 

6363  1.5717 

6610  1.5127 

6864  1.4568 

32 

29 

5887  1.6988 

6124  1.6329 

6367  1.5707 

6615  1.5118 

6869  1.4559 

31 

30 

5890  1.6977 

6128  1.6319 

6371  1.5697 

6619  1.5108 

6873  1.4550 

30 

31 

5894  1.6965 

6132  1.6308 

6375  1.5687 

6623  1.5099 

6877  1.4541 

29 

32 

5898  1.6954 

6136  1.6297 

6379  1.5677 

6627  1.5089 

6881  1.4532 

28 

33 

5902  1.6943 

6140  1.6287 

6383  1.5667 

6631  1.5080 

6886  1.4523 

27 

34 

5906  1.6932 

6144  1.6276 

6387  1.5657 

6636  1.5070 

6890  1.4514 

26 

35 

5910  1.6920 

6148  1.6265 

6391  1.5647 

6640  1.5061 

6894  1.4505 

25 

36 

5914  1.6909 

6152  1.6255 

6395  1.5637 

6644  1.5051 

6899  1.4496 

24 

37 

5918  1.6898 

6156  1.6244 

6399  1.5627 

6648  1.5042 

6903  1.4487 

23 

38 

5922  1.6887 

6160  1.6234 

6403  1.5617 

6652  1.5032 

6907  1.4478 

22 

39 

5926  1.6875 

6164  1.6223 

6408  1.5607 

6657  1.5023 

6911  1.4469 

21 

4O 

5930  1.6864 

6168  1.6212 

6412  1.5597 

6661  1.5013 

6916  1.4460 

20 

41 

5934  1.6853 

6172  1.6202 

6416  1.5587 

6665  1.5004 

6920  1.4451 

19 

42 

5938  1.6842 

6176  1.6191 

6420  1.5577 

6669  1.4994 

6924  1.4442 

18 

43 

5942  1.6831 

6180  1.6181 

6424  1.5567 

6673  1.4985 

6929  1.4433 

17 

44 

5945  1.6820 

6184  1.6170 

6428  1.5557 

6678  1.4975 

6933  1.4424 

16 

45 

5949  1.6808 

6188  1.6160 

6432  1.5547 

6682  1.4966 

6937  1.4415 

15 

46 

5953  1.6797 

6192  1.6149 

6436  1.5537 

6686  1.4957 

6942  1.4406 

14 

47 

5957  1.6786 

6196  1.6139 

6440  1.5527 

6690  1.4947 

6946  1.4397 

13 

48 

5961  1.6775 

6200  1.6128 

6445  1.5517 

6694  1.4938 

6950  1.4388 

12 

49 

5965  1.6764 

6204  1.6118 

6449  1.5507 

6699  1.4928 

6954  1.4379 

11 

50 

5969  1.6753 

6208  1.6107 

6453  1.5497 

6703  1.4919 

6959  1.4370 

10 

51 

5973  1.6742 

6212  1.6097 

6457  1.5487 

6707  1.4910 

6963  1.4361 

9 

52 

5977  1.6731 

6216  1.6087 

6461  1.5477 

6711  1.4900 

6967  1.4352 

8 

53 

5981  1.6720 

6220  1.6076 

6465  1.5468 

6716  1.4891 

6972  1.4344 

7 

54 

5985  1.6709 

6224  1.6066 

6469  1.5458 

6720  1.4882 

6976  1.4335 

6 

55 

5989  1.6698 

6228  1.6055 

6473  1.5448 

6724  1.4872 

6980  1.4326 

5 

56 

5993  1.6687 

6233  1.6045 

6478  1.5438 

6728  1.4863 

6985  1.4317 

4 

57 

5997  1.6676 

6237  1.6034 

6482  1.5428 

6732  1.4854 

6989  1.4308 

3 

58 

6001  1.6665 

6241  1.6024 

6486  1.5418 

6737  1.4844 

6993  1.4299 

2 

59 

6005  1.6654 

6245  1.6014 

6490  1.5408 

6741  1.4835 

6998  1.4290 

1 

60 

6009  1.6643 

6249  1.6003 

6494  1.5399 

6745  1.4826 

7002  1.4281 

0 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot  tan 

t 

59° 

68° 

57° 

56° 

55° 

t 

68 


NATURAL  TANGENTS   AND   COTANGENTS. 


f 

35° 

36° 

37° 

38° 

39° 

t 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

0 

7002  1.4281 

7265  1.3764 

7536  1.3270 

7813  1.2799 

8098  1.2349 

6O 

1 

7006  1.4273 

7270  1.3755 

7540  1.3262 

7818  1.2792 

8103  1.2342 

59 

2 

7011  1.4264 

7274  1.3747 

7545  1.3254 

7822  1.2784 

8107  1.2334 

58 

3 

7015  1.4255 

7279  1.3739 

7549  1.3246 

7827  1.2776 

8112  1.2327 

57 

4 

7019  1.4246 

7283  1.3730 

7554  1.3238 

7832  1.2769 

8117  1.2320 

56 

5 

7024  1.4237 

7288  1.3722 

7558  1.3230 

7836  1.2761 

8122  1.2312 

55 

6 

7028  1.4229 

7292  1.3713 

7563  1.3222 

7841  1.2753 

8127  1.2305 

54 

7 

7032  1.4220 

7297  1.3705 

7568  1.3214 

7846  1.2746 

8132  1.2298 

53 

8 

7037  1.4211 

7301  1.3697 

7572  1.3206 

7850  1.2738 

8136  1.2290 

52 

9 

7041  1.4202 

7306  1.3688 

7577  1.3198 

7855  1.2731 

8141  1.2283 

51 

1O 

7046  1.4193 

7310  1.3680 

7581  1.3190 

7860  1.2723 

8146  1.2276 

50 

11 

7050  1.4185 

7314  1.3672 

7586  1.3182 

7865  1.2715 

8151  1.2268 

49 

12 

7054  1.4176 

7319  1.3663 

7590  1.3175 

7869  1.2708 

8156  1.2261  . 

48 

13 

7059  1.4167 

7323  1.3655 

7595  1.3167 

7874  1.2700 

8161  1.2254 

47 

14 

7063  1.4158 

7328  1.3647 

7600  1.3159 

7879  1.2693 

8165  1.2247 

46 

15 

7067  1.4150 

7332  1.3638 

7604  1.3151 

7883  1.2685 

8170  1.2239 

45 

16 

7072  1.4141 

7337  1.3630 

7609  1.3143 

7888  1.2677 

8175  1.2232 

44 

17 

7076  1.4132 

7341  1.3622 

7613  1.3135 

7893  1.2670 

8180  1.2225 

43 

18 

7080  1.4124 

7346  1.3613 

7618  1.3127 

7898  1.2662 

8185  1.2218 

42 

19 

7085  1.4115 

7350  1.3605 

7623  1.3119 

7902  1.2655 

8190  1.2210 

41 

2O 

7089  1.4106 

7355  1.3597 

7627  1.3111 

7907  1.2647 

8195  1.2203 

4O 

21 

7094  1.4097 

7359  1.3588 

7632  1.3103 

7912  1.2640 

8199  1.2196 

39 

22 

7098  1.4089 

7364  1.3580 

7636  1.3095 

7916  1.2632 

8204  1.2189 

38 

23 

7102  1.4080 

7368  1.3572 

7641  1.3087 

7921  1.2624 

8209  1.2181 

37 

24 

7107  1.4071 

7373  1.3564 

7646  1.3079 

7926  1.2617 

8214  1.2174 

36 

25 

7111  1.4063 

7377  1.3555 

7650  1.3072 

7931  1.2609 

8219  1.2167 

35 

26 

'7115  1.4054 

7382  1.3547 

7655  1.3064 

7935  1.2602 

8224  1.2160 

34 

27 

7120  1.4045 

7386  1.3539 

7659  1.3056 

7940  1.2594 

8229  1.2153 

33 

28 

7124  1.4037 

7391  1.3531 

7664  1.3048 

7945  1.2587 

8234  1.2145 

32 

29 

7129  1.4028 

7395  1.3522 

7669  1.3040 

7950  1.2579 

8238  1.2138 

31 

30 

7133  1.4019 

7400  1.3514 

7673  1.3032 

7954  1.2572 

8243  1.2131 

30 

31 

7137  1.4011 

7404  1.3506 

7678  1.3024 

7959  1.2564 

8248  1.2124 

29 

32 

7142  1.4002 

7409  1.3498 

7683  1.3017 

7964  1.2557 

8253  1.2117 

28 

33 

7146  1.3994 

7413  1.3490 

7687  1.3009 

7969  1.2549 

8258  1.2109 

27 

34 

7151  1.3985 

7418  1.3481 

7692  1.3001 

7973  1.2542 

8263  1.2102 

26 

35 

7155  1.3976 

7422  1.3473 

7696  1.2993 

7978  1.2534 

8268  1.2095 

25 

36 

7159  1.3968 

7427  1.3465 

7701  1.2985 

7983  1.2527 

8273  1.2088 

24 

37 

7164  1.3959 

7431  1.3457 

7706  1.2977 

7988  1.2519 

8278  1.2081 

23 

38 

7168  1.3951 

7436  1.3449 

7710  1.2970 

7992  1.2512 

8283  1.2074 

22 

39 

7173  1.3942 

7440  1.3440 

7715  1.2962 

7997  1.2504 

8287  1.2066 

21 

4O 

7177  1.3934 

7445  1.3432 

7720  1.2954 

8002  1.2497 

8292  1.2059 

2O 

41 

7181  1.3925 

7449  1.3424 

7724  1.2946 

8007  1.2489 

8297  1.2052 

19 

42 

7186  1.3916 

7454  1.3416 

7729  1.2938 

8012  1.2482 

8302  1.2045 

18 

43 

7190  1.3908 

7458  1.3408 

7734  1.2931 

8016  1.2475 

8307  1.2038 

17 

44 

7195  1.3899 

7463  1.3400 

7738  1.2923 

8021  1.2467 

8312  1.2031 

16 

45 

7199  1.3891 

7467  1.3392 

7743  1.2915 

8026  1.2460 

8317  1.2024 

15 

46 

7203  1.3882 

7472  1.3384 

7747  1.2907 

8031  1.2452 

8322  1.2017 

14 

47 

7208  1.3874 

7476  1.3375 

7752  1.2900 

8035  1.2445 

8327  1.2009 

13 

48 

7212  1.3865 

7481  1.3367 

7757  1.2892 

8040  1.2437 

8332  1.2002 

12 

49 

7217  1.3857 

7485  1.3359 

7761  1.2884 

8045  1.2430 

8337  1.1995 

11 

5O 

7221  1.3848 

7490  1.3351 

7766  1.2876 

8050  1.2423 

8342  1.1988 

10 

51 

7226  1.3840 

7495  1.3343 

7771  1.2869 

8055  1.2415 

8346  1.1981 

9 

52 

7230  1.3831 

7499  1.3335 

7775  1.2861 

8059  1.2408 

8351  1.1974 

8 

53 

7234  1.3823 

7504  1.3327 

7780  1.2853 

8064  1.2401 

8356  1.1967 

7 

54 

7239  1.3814 

7508  1.3319 

7785  1.2846 

8069  1.2393 

8361  1.1960 

6 

55 

7243  1.3806 

7513  1.3311 

7789  1.2838 

8074  1.2386 

8366  1.1953 

5 

56 

7248  1.3798 

7517  1.3303 

7794  1.2830 

8079  1.2378 

8371  .1.1946 

4 

57 

7252  1.3789 

7522  1.3295 

7799  1.2822 

8083  1.2371 

8376  1.1939 

3 

58 

7257  1.3781 

7526  1.3287 

7803  1.2815 

8088  1.2364 

8381  1.1932 

2 

59 

7261  1.3772 

7531  1.3278 

7808  1.2807 

8093  1.2356 

8386  1.1925 

1 

6O 

7265  1.3764 

7536  1.3270 

7813  1.2799 

8098  1.2349 

8391  1.1918 

O 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

t 

54° 

53° 

52° 

51° 

5O° 

r 

NATURAL  TANGENTS   AND   COTANGENTS. 


69 


t 

40° 

41° 

42° 

43° 

44° 

t 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

tan   cot 

o 

8391  1.1918 

8693  1.1504 

9004  1.1106 

9325  1.0724 

9657  1.0355 

60 

1 

8396  1.1910 

8698  1.1497 

9009  1.1100 

9331  1.0717 

9663  1.0349 

59 

2 

8401  1.1903 

8703  1.1490 

9015  1.1093 

9336  1.0711 

9668  1.0343 

58 

3 

8406  1.1896 

8708  1.1483 

9020  1.1087 

9341  1.0705 

9674  1.0337 

57 

4 

8411  1.1889 

8713  1.1477 

9025  1.1080 

9347  1.0699 

9679  1.0331 

56 

5 

8416  1.1882 

8718  1.1470 

9030  1.1074 

9352  1.0692 

9685  1.0325 

55 

6 

8421  1.1875 

8724  1.1463 

9036  1.1067 

9358  1.0686 

9691  1.0319 

54 

7 

8426  1.1868 

8729  1.1456 

9041  1.1061 

9363  1.0680 

9696  1.0313 

53 

8 

8431  1.1861 

8734  1.1450 

9046  1.1054 

9369  1.0674 

9702  1.0307 

52 

9 

8436  1.1854 

8739  1.1443 

9052  1.1048 

9374  1.0668 

9708  1.0301 

51 

10 

8441  1.1847 

8744  1.1436 

9057  1.1041 

9380  1.0661 

9713  1.0295 

50 

11 

8446  1.1840 

8749  1.1430 

9062  1.1035 

9385  1.0655 

9719  1.0289 

49 

12 

8451  1.1833 

8754  1.1423 

9067  1.1028 

9391  1.0649 

9725  1.0283 

48 

13 

8456  1.1826 

8759  1.1416 

9073  1.1022 

9396  1.0643 

9730  1.0277 

47 

14 

8461  1.1819 

8765  1.1410 

9078  1.1016 

9402  1.0637 

9736  1.0271 

46 

15 

8466  1.1812 

8770  1.1403 

9083  1.1009 

9407  1.0630 

9742  1.0265 

45 

16 

8471  1.1806 

8775  1.1396 

9089  1.1003 

9413  1.0624 

9747  1.0259 

44 

17 

8476  1.1799 

8780  1.1389 

9094  1.0996 

9418  1.0618 

9753  1.0253 

43 

18 

8481  1.1792 

8785  1.1383. 

9099  1.0990 

9424  1.0612 

9759  1.0247 

42 

19 

8486  1.1785 

8790  1.1376 

9105  1.0983 

9429  1.0606 

9764  1.0241 

41 

2O 

8491  1.1778 

8796  1.1369 

9110  1.0977 

9435  1.0599 

9770  1.0235 

40 

21 

8496  1.1771 

8801  1.1363 

9115  1.0971 

9440  1.0593 

9776  1.0230 

39 

22 

8501  1.1764 

8806  1.1356 

9121  1.0964 

9446  1.0587 

9781  1.0224 

38 

23 

8506  1.1757 

8811  1.1349 

9126  1.0958 

9451  1.0581 

9787  1.0218 

37 

24 

8511  1.1750 

8816  1.1343 

9131  1.0951 

9457  1.0575 

9793  1.0212 

36 

25 

8516  1.1743 

8821  1.1336 

9137  1.0945 

9462  1.0569 

9798  1.0206 

35 

26 

8521  1.1736 

8827  1.1329 

9142  1.0939 

9468  1.0562 

9804  1.0200 

34 

27 

8526  1.1729 

8832  1.1323 

9147  1.0932 

9473  1.0556 

9810  1.0194 

33 

28 

8531  1.1722 

8837  M316 

9153  1.0926 

9479  1.0550 

9816  1.0188- 

32 

29 

8536  1.1715 

8842  1.1310 

9158  1.0919 

9484  1.0544 

9821  1.0182 

31 

3D 

8541  1.1708 

8847  1.1303 

9163  1.0913 

9490  1.0538 

9827  1.0176 

30 

31 

8546  1.1702 

8852  1.1296 

9169  1.0907 

9495  1.0532 

9833  1.0170 

29 

32 

8551  1.1695 

8858  1.1290 

9174  1.0900 

9501  1.0526 

9838  1.0164 

28 

33 

8556  1.1688 

8863  1.1283 

9179  1.0894 

9506  1.0519 

9844  1.0158 

27 

34 

8561  1.1681 

8868  1.1276 

9185  1.0888 

9512  1.0513 

9850  1.0152 

26 

35 

8566  1.1674 

8873  1.1270 

9190  1.0881 

9517  1.0507 

9856  1.0147 

25 

36 

8571  1.1667 

8878  1.1263 

9195  1.0875 

9523  1.0501 

9861  1.0141 

24 

37 

8576  1.1660 

8884  1.1257 

9201  1.0869 

9528  1.0495 

9867  1.0135 

23 

38 

8581  1.1653 

8889  1.1250 

9206  1.0862 

9534  1.0489 

9873  1.0129 

22 

39 

8586  1.1647 

8894  1.1243 

9212  1.0856 

9540  1.0483 

9879  1.0123 

21 

4O 

8591  1.1640 

8899  1.1237 

9217  1.0850 

9545  1.0477 

9884  1.0117 

2O 

41 

8596  1.1633 

8904  1.1230 

9222  1.0843 

9551  1.0470 

9890  1.0111 

19 

42 

8601  1.1626 

8910  1.1224 

9228  1.0837 

9556  1.0464 

9896  1.0105 

18 

43 

8606  1.1619 

8915  1.1217 

9233  1.0831 

9562  1.0458 

9902  1.0099 

17 

44 

8611  1.1612 

8920  1.1211 

9239  1.0824 

9567  1.0452 

9907  1.0094 

16 

45 

8617  1.1606 

8925  1.1204 

9244  1.0818 

9573  1.0446 

9913  1.0088 

15 

46 

8622  1.1599 

8931  1.1197 

9249  1.0812 

9578  1.0440 

9919  1.0082 

14 

47 

8627  1.1592 

8936  1.1191 

9255  1.0805 

9584  1.0434 

9925  1.0076 

13 

48 

8632  1.1585 

8941  1.1184 

9260  .1.0799 

9590  1.0428 

9930  1.0070 

12 

49 

8637  1.1578 

8946  1.1178 

9266  1.0793 

9595  1.0422 

9936  1.0064 

11 

50 

8642  1.1571 

8952  1.1171 

9271  1.0786 

9601  1.0416 

9942  1.0058 

1O 

51 

8647  1.1565 

8957  1.1165 

9276  1.0780 

9606  1.0410 

9948  1.0052 

9 

52 

8652  1.1558 

8962  1.1158 

9282  1.0774 

9612  1.0404 

9954  1.0047 

8 

53 

8657  1.1551 

8967  1.1152 

9287  1.0768 

9618  1.0398 

9959  1.0041 

7 

54 

8662  1.1544 

8972  1.1145 

9293  1.0761 

9623  1.0392 

9965  1.0035 

6 

55 

8667  1.1538 

8978  1.1139 

9298  1.0755 

9629  1.0385 

9971  1.0029 

5 

56 

8672  1.1531 

8983  1.1132 

9303  1.0749 

9634  1.0379 

9977  1.0023 

4 

57 

,  8678  1.1524 

8988  1.1126 

9309  1.0742 

9640  1.0373 

9983  1.0017 

3 

58 

8683  1.1517 

8994  1.1119 

9314  1.0736 

9646  1.0367 

9988  1.0012 

2 

59 

8688  1.1510 

8999  1.1113 

9320  1.0730 

9651  1.0361 

9994  1.0006 

1 

60 

8693  1.1504 

9004  1.1106 

9325  1.0724 

9657  1.0355 

1.000  1.0000 

O 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

cot   tan 

t 

49° 

48° 

47° 

46° 

45° 

/ 

70 


TABLE  VII.-TRAVEKSE  TABLE. 


Bearing, 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

0       f 

Lat,      Dep, 

Lat.      Dep, 

Lat,      Dep, 

Lat,      Dep, 

Lat,      Dep, 

o     / 

CIS 

1.000  0.004 

2.000  0.009 

3.000  0.013 

4.000  0.017 

5.000  0.022 

8945 

30 

1.000  0.009 

2.000  0.017 

3.000  0.026 

4.000  0.035 

5.000  0.044 

30 

45 

1.000  0.013 

2.000  0.026 

3.000  0.039 

4.000  0.052 

5.000  0.065 

15 

1    0 

1.000   0.017 

2.000  0.035 

3.000  0.052 

3.999  0.070 

4.999  0.087 

89   0 

15 

1.000  0.022 

2.000  0.044 

2.999  0.065 

3.999  0.087 

4.999  0.109 

45 

30 

1.000  0.026 

1.999  0.052 

2.999  0.079 

3.999  0.105 

4.998  0.131 

30 

45 

1.000  0.031 

1.999  0.061 

2.999   0.092 

3.998  0.122 

4.998  0.153 

15 

2    0 

0.999  0.035 

1.999  0.070 

2.998   0.105 

3.998  0.140 

4.997  0.174 

88   0 

15 

0.999  0.039 

1.998   0.079 

2.998  0.118 

3.997  0.157 

4.996  0.196 

45 

30 

0.999  0.044 

1.998  0.087 

2.997   0.131 

3.996  0.174 

4.995   0.218 

30 

45 

0.999  0.048 

1.998   0.096 

2.997   0.144 

3.995   0.192 

4.994  0.240 

15 

3    0 

0.999  0.052 

1.997   0.105 

2.996  0.157 

3.995   0.209 

4.993   0.262 

87    0 

15 

0.998  0.057 

1.997  0.113 

2.995   0.170 

3.994  0.227 

4.992  0283 

45 

30 

0.998  0.061 

1.996  0.122 

2.994   0.183 

3.993   0.244 

4.991   0.305 

30 

45 

0.998  0.065 

1.996  0.131 

2.994   0.196 

3.991   0.262 

4.989  0.327 

15 

40 

0.998  0070 

1.995   0.140 

2.993   0.209 

3.990  0.279 

4.988  0349 

86   0 

15 

0.997   0.074 

1.995   0.148 

2.992   0.222 

3.989  0.296 

4.986  0.371 

45 

30 

0.997  0.078 

1.994  0.157 

2.991    0.235 

3.988  0.314 

4.985   0.392 

30 

45 

0.997  0.083 

1.993   0.166 

2.990   0.248 

3.986  0.331 

4.983   0.414 

15 

5   0 

0.996  0.087 

1.992  0.174 

2.989   0.261 

3.985   0.349 

4.981  0.436 

85    0 

15 

0.996  0.092 

1.992   0.183 

2.987   0.275 

3.983  0.366 

4.979  0.458 

45 

30 

0.995   0096 

1.991   0192 

2.986   0.288 

3.982  0.383 

4.977  0.479 

30 

45 

0.995   0.100 

1.990  0.200 

2.985    0.301 

3.980  0.401 

4.975   0.501 

15 

6   0 

0.995   0.105 

1.989  0209 

2.984   0.314 

3.978  0.418 

4.973   0.523 

84    0 

15 

0.994  0.109 

1.988  0.218 

2.982   0.327 

3.976  0.435 

4.970  0.544 

45 

30 

0.994  0.113 

1.987  0.226 

2.981    0.340 

3.974  0.453 

4.968  0.566 

30 

45 

0.993   0.118 

1.986  0.235 

2.979   0.353 

3.972  0.470 

4.965   0.588 

15 

7    0. 

0.993   0.122 

1.985   0.244 

2.978   0.366 

3.970  0.487 

4.963   0.609 

83   0 

15 

0.992   0.126 

1.984  0.252 

2.976  0.379 

3.968  0.505 

4.960  0.631 

45 

30 

0.991   0.131 

1.983   0.261 

2.974   0.392 

3.966  0.522 

4.957  0.653 

30 

45 

0.991   0.135 

1.982  0.270 

2.973   0.405 

3.963  0.539 

4.954  0.674 

15 

8   0 

0.990  0.139 

1.981   0.278 

2.971    0.418 

3.961   0.557 

4.951   0.696 

82    0 

15 

0.990  0.143 

1.979  0.287 

2.969   0.430 

3.959  0.574 

4.948  0.717 

45 

30 

0.989  0.148 

1.978  0.296 

2.967   0.443 

3.956  0.591 

4.945   0.739 

30 

45 

0.988  0.152 

1.977   0.304 

2.965   0.456 

3.953  0.608 

4.942  0.761 

15 

9    0 

0.988  0.156 

1.975   0.313 

2.963   0.469 

3.951   0.626 

4.938  0.782 

81    0 

15 

0.987  0.161 

1.974  0.321 

2.961   0.482 

3.948  0.643 

4.935   0.804 

45 

30 

0.986  0.165 

1.973   0.330 

2.959  0.495 

3.945   0.660 

4.931   0.825 

30 

45 

0.986  0.169 

1.971   0.339 

2.957  0.508 

3.942  0.677 

4.928  0.847 

15 

1O    0 

0.985  0.174 

1.970  0.347 

2.954   0.521 

3.939  0.695 

4.924  0.868 

8O    0 

15 

0984  0.178 

1.968  0.356 

2.952   0.534 

3.936  0.712 

4.920  0.890 

45 

30 

0.983  0.182 

1.967  0.364 

2.950  0.547 

3.933   0.729 

4.916  0.911 

30 

45 

0.982  0.187 

1.965  0.373 

2.947   0.560 

3.930  0.746 

4.912  0.933 

15 

11    0 

0.98?,  0.191 

1.963  0.382 

2.945   0.572 

3.927  0.763 

4.908   0.954 

79   0 

15 

0.981   0.195 

1.962  0.390 

2.942   0.585 

3.923  0.780 

4.904  0.975 

45 

30 

0.980  0.199 

1.960  0.399 

2.940   0.598 

3.920  0.797 

4.900  0.997 

30 

45 

0.979  0.204 

1.958  0.407 

2.937  0611 

3.916  0.815 

4.895    1.018 

15 

12    0 

0.978  0.208 

1.956  0.416 

2.934  "0624 

3.913  0.832 

4.891    1.040 

78    0 

15 

0.977  0.212 

1.954  0.424 

2.932  0.637 

3.909  0.849 

4886   1.061 

45 

30 

0.976  0.216 

1.953   0.433 

2.929  0.649 

3.905   0.866 

4.881    1.082 

30 

45 

0.975   0.221 

1.951   0.441 

2.926  0662 

3.901   0883 

4.877   1.103 

15 

13   0 

0.974  0.225 

1.949  0.450 

2.923  0.675 

3.897   0.900 

4.872   1.125 

77    0 

15 

0.973   0.229 

1.947  0.458 

2.920  0.688 

3.894  0.917 

4.867   1.146 

45 

30 

0.972  0.233 

1.945   0.467 

2.917  0.700 

3.889  0.934 

4.862   1.167 

30 

45 

0971   0.238 

1.943  0.475 

2.914  0.713 

3.885  0.951 

4.857   1.188 

15 

14   0 

0.970  0.242 

1.941   0.484 

2.911   0.726 

3.881  0.968 

4.851    1.210 

76   0 

15 

0.969  0.246 

1.938  0.492 

2.908  0.738 

3.877  0.985 

4.846   1.231 

45 

30 

0.968  0.250 

1.936  0.501 

2.904  0.751 

3.873   1.002 

4.841    1.252 

30 

45 

0.967  0.255 

1.934  0.509 

2.901   0.764 

3.868   1.018 

4.835    1.273 

15 

15    0 

0.966  0.259 

1.932  0.518 

2.898  0.776 

3.864  1.035 

4.830   1.294 

75   C 

0       f 

Dep,     Lat, 

Dep,       Lat, 

Dep,      Lat, 

Dep,     Lat, 

Dep.     Lat, 

o     t 

Bearing, 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing, 

75°-90< 


71 


Bearing, 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing, 

o    r 

Lat, 

Dep, 

Lat, 

Dep, 

Lat, 

Dep, 

Lat. 

Dep, 

Lat, 

Dep, 

o     r 

O15 

6.000 

0.026 

7.000 

0.031 

8.000 

0.035 

9.000 

0.039 

10.000 

0.044 

8945 

30 

6.000 

0.052 

7.000 

0.061 

8.000 

0.070 

9.000 

0.079 

10.000 

0.087 

30 

45 

5.999 

0.079 

6.999 

0.092 

7.999 

0.105 

8.999 

0.118 

9.999 

0.131 

15 

1    0 

5.999 

0.105 

6.999 

0.122 

7.999 

0.140 

8.999 

0.157 

9.999 

0.175 

89   0 

15 

5.999 

0.131 

6.998 

0.153 

7.998 

0.175 

8.998 

0.196 

9.998 

0.218 

45 

30 

5.998 

0.157 

6.998 

0.183 

7.997 

0.209 

8.997 

0.236 

9.997 

0.262 

30 

45 

5.997 

0.183 

6.997 

0.214 

7.996 

0.244 

8.996 

0.275 

9.995 

0.305 

15 

2    0 

5.996 

0.209 

6.996 

0.244 

7.995 

0.279 

8.995 

0.314 

9.994 

0.349 

88    0 

15 

5.995 

0.236 

6.995 

0.275 

7.994 

0.314 

8.993 

0.353 

9.992 

0.393 

45 

30 

5.994 

0.262 

6.993 

0.305 

7.992 

0.349 

8.991 

0.393 

9.991 

0.436 

30 

45 

5.993 

0.288 

6.992 

0.336 

7.991 

0.384 

8.990 

0.432 

9.989 

0.480 

15 

3   0 

5.992 

0.314 

6.990 

0.366 

7.989 

0.419 

8.988 

0.471 

"   9.986 

0.523 

87    0 

15 

5.990 

0.340 

6.989 

0.397 

7.987 

0.454 

8.986 

0.510 

9.984 

0.567 

45 

30 

5.989 

0.366 

6.987 

0.427 

7.985 

0.488 

8.983 

0.549 

9.981 

0.611 

30 

45 

5.987 

0.392 

6.985 

0.458 

7.983 

0.523 

8.981 

0.589 

9.979 

0.654 

15 

4   0 

5.985 

0.419 

6.983 

0.488 

7.981 

0.558 

8.978 

0.628 

9.976 

0.698 

86   0 

15 

5.984 

0.445 

6.981 

0.519 

7.978 

0.593 

8.975 

0.667 

9.973 

0.741 

45 

30 

5.982 

0.471 

6.978 

0.549 

7.975 

0.628 

8.972 

0.706 

9.969 

0.785 

30 

45 

5.979 

0.497 

6.976 

0.580 

7.973 

0.662 

8.969 

0.745 

9.966 

0.828 

15 

5    0 

5.977 

0.523 

6.973 

0.610 

7.970 

0.697 

8.966 

0.784 

9.962 

0.872 

85    0 

15 

5.975 

0.549 

6.971 

0.641 

7.966 

0.732 

8.962 

0.824 

9.958 

0.915 

45 

30 

5.972 

0.575 

6.968 

0.671 

7.963 

0.767 

8.959 

0.863 

9.954 

0.959 

30 

45 

5.970 

0.601 

6.965 

0.701 

7.960 

0.802 

8.955 

0.902 

9.950 

1.002 

15 

6    0 

5.967 

0.627 

6.962 

0.732 

7.956 

0.836 

8.951 

0.941 

9.945 

1.045 

84    0 

15 

5.964 

0.653 

6.958 

0.762 

7.952 

0.871 

8.947 

0.980 

9.941 

1.089 

45 

30 

5.961 

0.679 

6.955 

0.792 

7.949 

0.906 

8.942 

1.019 

9.936 

1.132 

30 

45 

5.958 

0.705 

6.951 

0.823 

7.945 

0.940 

8.938 

1.058 

9.931 

1.175 

15 

7    0 

5.955 

0.731 

6.948 

0.853 

7.940 

0.975 

8.933 

1.097 

9.926 

1.219 

83    0 

15 

5.952 

0.757 

6.944 

0.883 

7.936 

1.010 

8.928 

1.136 

9.920 

1.262 

45 

30 

5.949 

0.783 

6.940 

0.914 

7.932 

1.044 

8.923 

1.175 

9.914 

1.305 

30 

45 

5.945 

0.809 

6.936 

0.944 

7.927 

1.079 

8.918 

1.214 

9.909 

1.349 

15 

8    0 

5.942 

0.835 

6.932 

0.974 

7.922 

1.113 

8.912 

1.253 

9.903 

1.392 

82    0 

15 

5.938 

0.861 

6.928 

1.004 

7.917 

1.148 

8.907 

1.291 

9.897 

1.435 

45 

30 

5.934 

0.887 

6.923 

1.035 

7.912 

1.182 

8.901 

1.330 

9.890 

1.478 

30 

45 

5.930 

0913 

6.919 

1.065 

7.907 

1.217 

8.895 

1.369 

9.884 

1.521 

15 

9   0 

5.926 

0.939 

6.914 

1.095 

7.902 

1.251 

8.889 

1.408 

9.877 

1.564 

81    0 

15 

5.922 

0.964 

6.909 

1.125 

7.896 

1.286 

8.883 

1.447 

9.870 

1.607 

45 

30 

5.918 

0.990 

6.904 

1.155 

7.890 

1.320 

8.877 

1.485 

9.863 

1.651 

30 

45 

5.913 

1.016 

6.899 

1.185 

7.884 

1.355 

8.870 

1.524 

9.856 

1.694 

15 

1O    0 

5.909 

1.042 

6.894 

1.216 

7.878 

1.389 

8.863 

1.563 

9.848 

1.737 

80   0 

15 

5.904 

1.068 

6.888 

1.246 

7.872 

1.424 

8.856 

1.601 

9.840 

1.779 

45 

30 

5.900 

1.093 

6.883 

1.276 

7.866 

1.458 

8.849 

1.640 

9.833 

1.822 

30 

45 

5.895 

1.119 

6.877 

1.306 

7.860 

1.492 

8.842 

1.679 

9.825 

1.865 

15 

11    0 

5.890 

1.145 

6.871 

1.336 

7.853 

1.526 

8.835 

1.717 

9.816 

1.908 

79   0 

15 

5.885 

1.171 

6.866 

1.366 

7.846 

1.561 

8.827 

1.756 

9.808 

1.951 

45 

30 

5.880 

1.196 

6.859 

1.396 

7.839 

1.595 

8.819 

1.794 

9.799 

1.994 

30 

45 

5.874 

1.222 

6.853 

1.425 

7.832 

1  .629 

8.811 

1.833 

9.791 

2.036 

15 

12    0 

5.869 

1.247 

6.847 

1.455 

7.825 

1.663 

8.803 

1.871 

9.782 

2.079 

78    0 

15 

5.863 

1.273 

6.841 

1.485 

7.818 

1.697 

8.795 

1.910 

9.772 

2.122 

45 

30 

5.858 

1.299 

6.834 

1.515 

7.810 

1.732 

8.787 

1.948 

9.763 

2.164 

30 

*      45 

5.852 

1.324 

6.827 

1.545 

7.803 

1.766 

8.778 

1.986 

9.753 

2.207 

15 

13    0 

5.846 

1.350 

6.821 

1.575 

7.795 

1.800 

8.769 

2.025 

9.744 

2.250 

77    0 

15 

5.840 

1.375 

6.814 

1.604 

7.787 

1.834 

8.760 

2.063 

9.734 

2.292 

45 

30 

5.834 

1.401 

6.807 

1.634 

7.779 

1.868 

8.751 

2.101 

9.724 

2.335 

30 

45 

5.828 

1.426 

6.799 

1.664 

7.771 

1.902 

8.742 

2.139 

9.713 

2.377 

15 

14    0 

5.822 

1.452 

6.792 

1.693 

7.762 

1.935 

8.733 

2.177 

9.703 

2.419 

76   0 

15 

5.815 

1.477 

6.785 

1.723 

7.754 

1.969 

8.723 

2.215 

9.692 

2.462 

45 

30 

5.809 

1.502 

6.777 

1.753 

7.745 

2.003 

8.713 

2.253 

9.682 

2.504 

30 

45 

5.802 

1.528 

6.769 

1.782 

7.736 

2.037 

8.703 

2.291 

9.671 

2.546 

15 

15    0 

5.796 

1.553 

6.761 

1.812 

7.727 

2.071 

8.693 

2.329 

9.659 

2.588 

75   0 

0       f 

Dep, 

Lat, 

Dep, 

Lat, 

Dep. 

Lat, 

Dep, 

Lat, 

Dep, 

Lat, 

0       f 

Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  10. 

Bearing. 

75°-90c 


72 


15°-30( 


Bearing. 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

o    r 

Lat.      Dep, 

Lat.      Dep. 

Lat.      Dep, 

Lat,      Dep, 

Lat.      Dep. 

0        f 

1515 

0.965   0.263 

1.930  0.526 

2.894  0.789 

3.859   1.052 

4.824   1.315 

7445 

30 

0.964  0.267 

1.927   0.534 

2.891   0.802 

3.855    1.069 

4.818   1.336 

30 

45 

0.962   0.271 

1.925   0.543 

2.887  0.814 

3.850   1.086 

4.812   1.357 

15 

16   0 

0.961   0.276 

1.923   0.551 

2.884   0.827 

3.845    1.103 

4.806   1.378 

74   0 

15 

0.960  0.280 

1.920  0.560 

2.880  0.839 

3.840   1.119 

4.800   1.399 

45 

30 

0.959  0.284 

1.918  0.568 

2.876  0.852 

3.835    1.136 

4.794   1.420 

30 

45 

0.958  0.288 

1.915   0.576 

2.873   0.865 

3.830   1.153 

4.788   1.441 

15 

17    0 

0.956  0.292 

1.913   0.585 

2.869  0.877 

3.825    1.169 

4.782   1.462 

73   0 

15 

0.955   0.297 

.910  0.593 

2.865    0.890 

3.820   1.186 

4.775   1.483 

45 

30 

0.954  0.301 

.907   0.601 

2.861   0.902 

3.815    1.203 

4.769   1.504 

30 

45 

0.952  0.305 

.905   0.610 

2.857   0.915 

3.810   1.220 

4.762   1.524 

15 

18    0 

0.951   0.309 

•1.902   0.618 

2.853    0.927 

3.804   1.236 

4.755   1.545 

72    0 

15 

0.950  0.313 

1.899  0.626 

2.849   0.939 

3.799   1.253 

4.748   1.566 

45 

30 

0.948  0.317 

•1.897   0.635 

2.845    0.952 

3.793    1.269 

4.742   1.587 

30 

45 

0.947  0.321 

1.894  0.643 

2.841    0.964 

3.788   1.286 

4.735    1.607 

15 

19    0 

0.946  0.326 

1.891   0.651 

2.837    0.977 

3.782    1.302 

4.728   1.628 

71    0 

15 

0.944  0.330 

1.888   0.659 

2.832   0.989 

3.776   1.319 

4.720   1.648 

45 

30 

0.943   0.334 

1.885   0.668 

2.828    1.001 

3.771    1.335 

4.713    1.669 

30 

45 

0.941   0.338 

1.882  0.676 

2.824    1.014 

3.765   1.352 

4.706   1.690 

15 

2O    0 

0.940  0.342 

1.879   0.684 

2.819    1.026 

3.759   1.368 

4.698   1.710 

7O   0 

15 

0.938  0.346 

1.876  0.692 

2.815    1.038 

3.753    1.384 

4.691    1.731 

45 

30 

0.937  0.350 

1.873   0.700 

2.810    1.051 

3.747    1.401 

4.683   1.751 

30 

45 

0.935   0.354 

1.870  0.709 

2.805    1.063 

3.741   1.417 

4.676   1.771 

15 

21    0 

0.934  0.358 

1.867   0.717 

2.801    1.075 

3.734   1.433 

4.668   1.792 

69   0 

15 

0.932  0.362 

1.864  0.725 

2.796    1.087 

3.728   1.450 

4.660   1.812 

45 

30 

0.930  0.367 

1.861   0.733 

2.791    1.100 

3.722   1.466 

4.652   1.833 

30 

45 

0.929  0.371 

1.858  0.741 

2.786    1.112 

3.715    1.482 

4.644   1.853 

15 

22    0 

0.927  0.375 

1.854  0.749 

2.782    1.124 

3.709   1.498 

4.636   1.873 

68   0 

15 

0.926  0.379 

1.851   0.757 

2.777    1.136 

3.702   1.515 

4.628   1.893 

45 

30 

0.924  0.383 

1.848  0.765 

2.772    1.148 

3.696   1.531 

4.619   1.913 

30 

45 

0.922  0.387 

1.844  0.773 

2.767    1.160 

3.689   1.547 

4.611    1.934 

15 

23   0 

0.921   0.391 

1.841   0.781 

2.762    1.172 

3.682   1.563 

4.603    1.954 

67    0 

15 

0.919  0.395 

1.838  0.789 

2.756    1.184 

3.675    1.579 

4.594   1.974 

45 

30 

0.917  0.399 

1.834  0.797 

2.751    1.196 

3.668   1.595 

4.585    1.994 

30 

45 

0.915   0.403 

1.831   0.805 

2.746    1.208 

3.661    1.611 

4.577   2.014 

15 

24   0 

0.914  0.407 

1.827   0.813 

2.741    1.220 

3.654   1.627 

4.568   2.034 

66   0 

15 

0.912   0.411 

1.824  0.821 

2.735    1.232 

3.647   1.643 

4.559  2.054 

45 

30 

0.910  0.415 

1.820  0.829 

2.730    1.244 

3.640   1.659 

4.550  2.073 

30 

45 

0.908   0.419 

1.816  0.837 

2.724    1.256 

3.633   1.675 

4.541   2.093 

15 

25    0 

0.906  0.423 

1.813  0.845 

2.719    1.268 

3.625    1.690 

4.532   2.113 

65    0 

15 

0.904  0.427 

1.809  0.853 

2.713    1.280 

3.618    1.706 

4.522   2.133 

45 

30 

0.903   0.431 

1.805   0.861 

2.708    1.292 

3.610   1.722 

4.513   2.153 

30 

45 

0.901   0.434 

1.801  0.869 

2.702    1.303 

3.603    1.738 

4.503   2.172 

15 

26    0 

0.899  0.438 

1.798  0.877 

2.696    1.315 

3.595    1.753 

4.494   2.192 

64   0 

15 

0.897   0.442 

1.794  0.885 

2.691    1.327 

3.587   1.769 

4.484  2.211 

45 

30 

0.895   0.446 

1.790  0892 

2.685    1.339 

3.580   1.785 

4.475   2.231 

30 

45 

0.893   0.450 

1.786  0.900 

2.679    1.350 

3.572   1.800 

4.465   2.250 

15 

27    0 

0.891   0.454 

1.782  0.908 

2.673    1.362 

3.564   1.816 

4.455   2.270 

63    0 

15 

0.889  0.458 

1.778  0.916 

2.667    1.374 

3.556   1.831 

4.445   2.289 

45 

30 

0.887   0.462 

1.774  0.923 

2.661    1.385 

3.548   1.847 

4.435    2.309 

30 

45 

0.885   0.466 

1.770  0.931 

2.655    1.397 

3.540    1.862 

4.425   2.328 

15* 

28    0 

0.883   0.469 

1.766  0.939 

2.649   1.408 

3.532   1.878 

4.415   2.347 

62    0 

15 

0.881   0.473 

1.762  0.947 

2.643    1.420 

3.524   1.893 

4.404   2.367 

45 

30 

0.879  0.477 

1.758  0.954 

2.636   1.431 

3.515    1.909 

4.394   2.386 

30 

45 

0.877   0.481 

1.753   0.962 

2.630   1.443 

3.507   1.924 

4.384   2.405 

15 

29    0 

0.875  0.485 

1.749  0.970 

2.624   1.454 

3.498   1.939 

4.373   2.424 

61    0 

15 

0.872  0.489 

1.745   0.977 

2.617   1.466 

3.490   1.954 

4.362   2.443 

45 

30 

0.870  0.492 

1.741  0.985 

2.611    1.477 

3.481    1.970 

4.352   2.462 

30 

45 

0.868  0.496 

1.736  0.992 

2.605    1.489 

3.473   1.985 

4.341    2.481 

15 

3O   0 

0.866  0.500 

1.732   1.000 

2.598   1.500 

3.464  2.000 

4.330   2.500 

6O    0 

o    r 

Dep.     Lat, 

Dep,      Lat. 

Dep,      Lat, 

Dep,     Lat, 

Dep.     Lat. 

0       f 

Bearing, 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing. 

60°-75< 


15° -30' 


73 


Bearing.     Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing, 

| 
o    r 

Lat,      Dep. 

Lat,      Dep, 

Lat,      Dep, 

Lat,      Dep, 

Lat,      Dep, 

0       t 

1515 

5.789  1.578 

6.754   1.841 

7.718   2.104 

8.683   2.367 

9.648   2.630 

7445 

30 

5.782   1.603 

6.745    1.871 

7.709   2.138 

8.673   2.405 

9.636   2.672 

30 

45 

5.775    1.629 

6.737   1.900 

7.700   2.172 

8.662   2.443 

9.625   2.714 

15 

16   0 

5.768   1.654 

6.729   1.929 

7.690   2.205 

8.651    2.481 

9.613   2.756 

74   0 

15 

5.760   1.679 

6.720   1.959 

7.580  2.239 

8.640  2.518 

9.601   2.798 

45 

30 

5.753   1.704 

6.712   1.988 

7.671   2.272 

8.629   2.556 

9.588   2.840 

30 

45 

5.745    1.729 

6.703   2.017 

7.661   2.306 

8.618  2.594 

9.576  2.882 

15 

17    0 

5.738   1.754 

6.694  2.047 

7.650   2.339 

8.607   2.631 

9.563   2.924 

73   0 

15 

5.730  1.779 

6.685   2.076 

7.640  2.372 

8.595    2.669 

9.550  2.965 

45 

30 

5.722   1.804 

6.676   2.105 

7.630  2.406 

8.583   2.706 

9.537  3.007 

30 

45 

5.714   1.829 

6.667  2.134 

7.619   2.439 

8.572   2.744 

9.524  3.049 

15 

18    0 

5.706   1.854 

6.657  2.163 

7.608   2.472 

8.560  2.781 

9.511   3.090 

72   0 

15 

5.698   1.879 

6.648   2.192 

7.598   2.505 

8.547   2.818 

9.497   3.132 

45 

30 

5.690   1.904 

6.638  2.221 

7.587   2.538 

8.535   2.856 

9.483  3.173 

30 

45 

5.682   1.929 

6.629  2.250 

7.575   2.572 

8.522   2.893 

9.469  3.214 

15 

19    0 

5.673   1.953 

6.619  2.279 

7.564   2.605 

8.510  2.930 

9.455   3.256 

71    0 

15 

5.665    1.978 

6.609   2.308 

7.553   2.638 

8.497   2.967 

9.441   3.297 

45 

30 

5.656  2.003 

6.598  2.337 

7.541    2.670 

8.484  3.004 

9.426  3.338 

30 

45 

5.647  2.028 

6.588  2.365 

7.529  2.703 

8.471   3.041 

9.412  3.379 

15 

2O    0 

5.638  2.052 

6.578  2.394 

7.518   2.736 

8.457  3.078 

9.397  3.420 

7O   0 

15 

5.629  2.077 

6.567   2.423 

7.506   2.769 

8.444  3.115 

9.382  3.461 

45 

30 

5.620  2.101 

6.557   2.451 

7.493    2.802 

8.430  3.152 

9.367  3.502 

30 

45 

5.611   2.126 

6.546  2.480 

7.481    2.834 

8.416  3.189 

9.351   3.543 

15 

21    0 

5.601   2.150 

6.535    2.509 

7.469    2.867 

8.402  3.225 

9.336  3.584 

69    0 

15 

5.592   2.175 

6.524  2.537 

7.456    2.900 

8.388  3.262 

9.320  3.624 

45 

30 

5.582   2.199 

6.513   2.566 

7.443    2.932 

8.374  3.299 

9.304  3.665 

30 

45 

5.573   2.223 

6.502   2.594 

7.430    2.964 

8.359  3.335 

9.288  3.706 

15 

22   0 

5.563   2.248 

6.490   2.622 

7.417    2.997 

8.345   3.371 

9.272  3.746 

68   0 

15 

5.553   2.272 

6.479   2.651 

7.404   3.029 

8.330  3.408 

9.255   3.787 

45 

30 

5.543   2.296 

6.467   2.679 

7.391    3.061 

8.315   3.444 

9.239  3.827 

30 

45 

5.533   2.320 

6.455   2.707 

7.378   3.094 

8.300  3.480 

9.222  3.867 

15 

23   0 

5.523   2.344 

6.444   2.735 

7.364   3.126 

8.285   3.517 

9.205  3.907 

67    0 

15 

5.513  2.368 

6.432   2.763 

7.350  3.158 

8.269  3.553 

9.188  3.947 

45 

30 

5.502   2.392 

6.419   2.791 

7,336  3.190 

8.254  3.589 

9.171   3.988 

30 

45 

5.492   2.416 

6.407   2.819 

7.322   3.222 

8.238  3.625 

9.153  4.028 

15 

24   0 

5.481   2.440 

6.395    2.847 

7.308   3.254 

8.222  3.661 

9.136  4.067 

66   0 

15 

5.471   2.464 

6.382   2.875 

7.294   3.286 

8.206  3.696 

9.118  4.107 

45 

30 

5.460  2.488 

6.370   2.903 

7.280   3.318 

8.190  3.732 

9.100  4.147 

30 

45 

5.449  2.512 

6.357  2.931 

7.265   3.349 

8.173  3.768 

9.081  4.187 

15 

25   0 

5.438  2.536 

6.344  2.958 

7.250  3.381 

8.157  3.804 

9.063   4.226 

65   0 

15 

5.427   2.559 

6.331   2.986 

7.236   3.413 

8.140  3.839 

9.045   4.266 

45 

30 

5.416  2.583 

6.318  3.014 

7.221    3.444 

8.123  3.875 

9.026  4.305 

30 

45 

5.404  2.607 

6.305   3.041 

7.206   3.476 

8.106  3.910 

9.007  4.345 

15 

26    0 

5.393   2.630 

6.292  3.069 

7.190   3.507 

8.089  3.945 

8.988  4.384 

64   0 

15 

5.381   2.654 

6.278  3.096 

7.175   3.538 

8.072   3.981 

8.969  4.423 

45 

30 

5.370  2.677 

6.265   3.123 

7.160  3.570 

8.054  4.016 

8.949  4.462 

30 

45 

5.358   2.701 

6.251   3.151 

7.144  3.601 

8.037   4.051 

8.930  4.501 

15 

27    0 

5.346  2.724 

6.237   3.17S 

7.128   3.632 

8.019  4.086 

8.910  4.540 

63    0 

15 

5.334  2.747 

6.223   3.205 

7.112   3.663 

8.001   4.121 

8.890  4.579 

45 

•       30 

5.322   2.770 

6.209  3.232 

/.096   3.694 

7.983  4.156 

8.870  4.618 

30 

45 

5.310   2.794 

6.195   3.259 

7.080  3.725 

7.965   4.190 

8.850  4.656 

15 

fc8    0 

•5.298   2.817 

6.181   3.286 

7.064  3.756 

7.947  4.225 

8.829  4.695 

62    0 

15 

5.285   2.840 

6.166  3.313 

7.047  3.787 

7.928  4.260 

S.809  4.733 

45 

30 

5.273   2.863 

6.152  3.340 

7.031   3.817 

7.909  4.294 

8.788  4.772 

30 

45 

5.260   2.886 

6.137  3.367 

7.014  3.848 

7.891   4.329 

8.767  4.810 

15 

29    0 

5.248   2.909 

6.122  3.394 

6.997  3.878 

7.872  4.363 

8.746  4.848 

61    0 

15 

5.235    2.932 

6.107  3.420 

6.980  3.909 

7.852  4.398 

8.725   4.886 

45 

30 

5.222   2.955 

6.093  3.447 

6.963   3.939 

7.833   4.432 

8.704  4.924 

30 

45 

5.209   2.977 

6.077   3.474 

6.946  3.970 

7.814  4.466 

8.682  4.962 

15 

3D   0 

5.196  3.000 

6.062  3.500 

6.928  4.000 

7.794  4.500 

8.66/)   5.000 

60   0 

0       f 

Dep,     Lat, 

Dep,      Lat. 

Dep,      Lat,. 

Dep,     Lat, 

Dep,     Lat, 

0       f 

Bearing, 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing, 

60°-75C 


74 


30°-45( 


Bearing, 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing, 

o     t 

Lat,      Dep, 

Lat.      Dep, 

Lat,      Dep, 

Lat,      Dep, 

Lat,      Dep, 

0        f 

3O15 

0.864  0.504 

1.728   1.008 

2.592    1.511 

3.455   2.015 

4.319   2.519 

5945 

30 

0.862  0.508 

1.723    1.015 

2.585    1.523 

3.447   2.030 

4.308  2.538 

30 

45 

0.859  0.511 

1.719   1.023 

2.578   1.534 

3.438  2.045 

4.297   2.556 

15 

31    0 

0.857  0.515 

1.714   1.030 

2.572   1.545 

3.429   2.060 

4.286  2.575 

59   0 

15 

0.855   0.519 

1.710   1.038 

2.565    1.556 

3.420   2.075 

4.275   2.594 

45 

30 

0.853   0.522 

1.705    1.045 

2.558   1.567 

3.411    2.090 

4.263   2.612 

30 

45 

0.850  0.526 

1.701    1.052 

2.551    1.579 

3.401   2.105 

4.252   2.631 

15 

32    0 

0.848  0.530 

1.696   1.060 

2.544   1.590 

3.392   2.120 

4.240  2.650 

58   0 

15 

0.846  0.534 

1.691    1.067 

2.537    1.601 

3.383   2.134 

4.229   2.668 

45 

30 

0.843   0.537 

1.687   1.075 

2.530   1.612 

3.374   2.149 

4.217   2.686 

30 

45 

0.841   0.541 

1.682   1.082 

2.523    1.623 

3.364   2.164 

4.205   2.705 

15 

33    0 

0.839  0.545 

1.677   1.089 

2.516    1.634 

3.355   2.179 

4.193   2.723 

57    0 

15 

0.836  0.548 

1.673   1.097 

2.509    1.645 

3.345    2.193 

4.181   2.741 

45 

30 

0.834  0.552 

1.668   1.104 

2.502    1.656 

3.336  2.208 

4.169  2.760 

30 

45 

0.831   0-.556 

1.663    1.111 

2.494    1.667 

3.326   2.222 

4.157  2.778 

15 

34    0 

0.829  0.559 

1.658   1.118 

2.487    1.678 

3.316   2.237 

4.145   2.796 

56   0 

15 

0.827  0.563 

1.653    1.126 

2.480    1.688 

3.306   2.251 

4.133   2.814 

45 

30 

0.824  0.566 

1.648   1.133 

2.472    1.699 

3.297   2.266 

4.121   2.832 

30 

45 

0.822   0.570 

1.643    1.140 

2.465    1.710 

3.287   2.280 

4.108  2.850 

15 

35    0 

0.819  0.574 

1.638   1.147 

2.457    1.721 

3.277   2.294 

4.096  2.868 

55    0 

15 

0.817  0.577 

1.633    1.154 

2.450   1.731 

3.267   2.309 

4.083   2.886 

45 

30 

0.814  0.581 

1.628   1.161 

2.442    1.742 

3.257   2.323 

4.071   2.904 

30 

45 

0.812   0.584 

1.623    1.168 

2.435    1.753 

3.246  2.337 

4.058   2.921 

15 

36   0 

0.809  0.588 

1.618   1.176 

2.427    1.763 

3.236  2.351 

4.045   2.939 

54   0 

15 

0.806  0.591 

1.613    1.183 

2.419   1.774 

3.226   2.365 

4.032   2.957 

45 

30 

0.804  0.595 

1.608   1.190 

2.412   1.784 

3.215    2.379 

4.019  2.974 

30 

45 

0.801   0.598 

1.603    1.197 

2.404   1.795 

3.205    2.393 

4.006  2.992 

15 

37    0 

0.799  0.602 

1.597   1.204 

2.396   1.805 

3.195   2.407 

3.993   3.009 

53   0 

15 

0.796  0.605 

1.592   1.211 

2.388    1.816 

3.184   2.421 

3.980  3.026 

45 

30 

0.793   0.609 

1.587   1.218 

2.380    1.826 

3.173   2.435 

3.967  3.044 

30 

45 

0.791   0.612 

1.581    1.224 

2.372    1.837 

3.163   2.449 

3.953   3.061 

15 

38    0 

0.788  0.616 

1.576   1.231 

2.364    1.847 

3.152   2.463 

3.940  3.078 

52    0 

15 

0.785   0.619 

1.571    1.238 

2.356   1.857 

3.141    2.476 

3.927  3.095 

45 

30 

0.783   0.623 

1.565    1.245 

2.348   1.868 

3.130   2.490 

3.913   3.113 

30 

45 

0.780  0.626 

1.560   1.252 

2.340   1.878 

3.120  2.504 

3.899  3.130 

.     15 

39    0 

0.777  0.629 

1.554   1.259 

2.331    1.888 

3.109   2.517 

3.886  3.147 

51    0 

15 

0.774  0.633 

1.549   1.265 

2.323    1.898 

3.098   2.531 

3.872  3.164 

45 

30 

0.772  0.636 

1.543    1.272 

2.315    1.908 

3.086   2.544 

3.858  3.180 

30 

45 

0.769  0.639 

1.538   1.279 

2.307   1.918 

3.075   2.558 

3.844  3.197 

15 

4O    0 

0.766  0.643 

1.532   1.286 

2.298    1.928 

3.064  2.571 

3.830  3.214 

50   0 

15 

0.763   0.646 

1.526   1.292 

2.290   1.938 

3.053   2.584 

3.816  3.231 

45 

30 

0.760  0.649 

1.521    1.299 

2.281    1.948 

3.042   2.598 

3.802  3.247 

30 

45 

0.758  0.653 

1.515    1.306 

2.273    1.958 

3.030   2.611 

3.788  3.264 

15 

41    0 

0.755   0.656 

1.509   1.312 

2.264   1.968 

3.019   2.624 

3.774  3.280 

49   0 

15 

0.752  0.659 

1.504   1.319 

2.256   1.978 

3.007  2.637 

3.759  3.297 

45 

30 

0.749  0.663 

1.498   1.325 

2.247    1.988 

2.996   2.650 

3.745  3.313 

30 

45 

0.746  0.666 

1.492   1.332 

2.238    1.998 

2.984   2.664 

3.730  3.329 

15 

42    0 

0.743   0.669 

1.486   1.338 

2.229   2.007 

2.973   2.677 

3.716  3.346 

48    0 

15 

0.740  0.672 

1.480   1.345 

2.221   2.017 

2.961   2.689 

3.701  3.362 

45 

30 

0.737  0.676 

1.475   1.351 

2.212   2.027 

2.949   2.702 

3.686  3.378 

30- 

45 

0.734  0.679 

1.469   1.358 

2.203   2.036 

2.937   2.715 

3.672   3.394 

15 

43   0 

0.731   0.682 

1.463    1.364 

2.194   2.046 

2.925   2.728 

3.657  3.410 

47    0 

15 

0.728  0.685 

1.457   1.370 

2.185   2.056 

2.913   2.741 

3.642  3.426 

45 

30 

0.725   0.688 

1.451   1.377 

2.176  2.065 

2.901   2.753 

3.627  3.442 

30 

45 

0.722  0.692 

1.445   1.383 

2.167   2.075 

2.889  2.766 

3.612  3.458 

15 

44    0 

0.719  0.695 

1.439  1.389 

2.158  2.084 

2.877   2.779 

3.597  3.473 

46   0 

15 

0.716  0.698 

1.433   1.396 

2.149.  2.093 

2.865   2.791 

3.582  3.489 

45 

30 

0.713  0.701 

1.427   1.402 

2.140  2.103 

2.853   2.804 

3.566  3.505 

30 

45 

0.710  0.704 

1.420   1.408 

2.131   2.112 

2.841   2.816 

3.551  3.520 

15 

45    0 

0.707  0707 

1.414   1.414 

2.121   2.121 

2.828  2.828 

3.536  3.536 

45   0 

0      f 

Dep,     Lat, 

Dep,      Lat, 

Dep,      Lat, 

Dep,     Lat, 

Dep,     Lat, 

0       f 

Bearing, 

Distance  1. 

Distance  2. 

Distance  3. 

Distance  4. 

Distance  5. 

Bearing, 

45° -60° 


30° -45°  - 


75 


Bearing. 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing, 

o    r 

Lat,      Dep, 

Lat,      Dep, 

Lat,      Dep, 

Lat.      Dep. 

Lat,      Dep, 

0        f 

3O15 

5.183   3.023 

6.047  3.526 

6.911   4.030 

7.775   4.534 

8.638  5.038 

5945 

30 

5.170  3.045 

6.031   3.553 

6.893   4.060 

7.755  4.568 

8.616  5.075 

30 

45 

5.156  3.068 

6.016  3.579 

6.875   4.090 

7.735   4.602 

8.594  5.113 

15 

31    0 

5.143   3.090 

6.000  3.605 

6.857   4.120 

7.715   4.635 

8.572  5.150 

59   0 

15 

5.129  3.113 

5.984   3.631 

6.839  4.150 

7.694  4.669 

8.549  5.188 

45 

30 

5.116  3.135 

5.968  3.657 

6.821   4.180 

7.674  4.702 

8.526  5.225 

30 

45 

5.102  3.157 

5.952   3.683 

6.803   4.210 

7.653   4.736 

8.504   5.262 

15 

32    0 

5.088  3.180 

5.936  3.709 

6.784   4.239 

7.632  4.769 

8.481   5.299 

58   0 

15 

5.074  3.202 

5.920  3.735 

6.766   4.269 

7.612  4.802 

8.457  5.336 

45 

30 

5.060  3.224 

5.904  3.761 

6.747   4.298 

7.591   4.836 

8.434  5.373 

30 

45 

5.046  3.246 

5.887  3.787 

6.728   4.328 

7.569  4.869 

8.410  5.410 

15 

33    0 

5.032   3.268 

5.871   3.812 

6.709   4.357 

7.548  4.902 

8.387  5.446 

57   0 

15 

5.018  3.290 

5.854  3.838 

6.690   4.386 

7.527  4.935 

8.363   5.483 

45 

30 

5.003   3.312 

5.837  3.864 

6.671    4.416 

7.505   4.967 

8.339  5.519 

30 

45 

4.989  3.333 

5.820  3.889 

6.652    4.445 

7.483   5.000 

8.315   5.556 

15 

34   0 

4.974  3.355 

5.803   3.914 

6.632    4.474 

7.461   5.033 

8.290  5.592 

56   0 

15 

4.960  3.377 

5.786  3.940 

6.613    4.502 

7.439  5.065 

8.266  5.628 

45 

,      30 

4.945   3.398 

5.769  3.965 

6.593   4.531 

7.417  5.098 

8.241   5.664 

30 

45 

4.930  3.420 

5.752  3.990 

6.573   4.560 

7.395   5.130 

8.217   5.700 

15 

35    0 

4.915   3.441 

5.734  4.015 

6.553   4.589 

7.372  5.162 

8.192  5.736 

55   0 

15 

4.900  3.463 

5.716  4.040 

6.533  4.617 

7.350  5.194 

8.166  5.772 

45 

30 

4.885  3.484 

5.699  4.065 

6.513  4.646 

7.327   5.226 

8.141   5.807 

30 

45 

4.869  3.505 

5.681   4.090 

6.493   4.674 

7.304   5.258 

8.116  5.843 

15 

36    0 

4.854  3.527 

5.663   4.115 

6.472  4.702 

7.281   5.290 

8.090  5.878 

54   0 

15 

4.839  3.548 

5.645   4.139 

6.452   4.730 

7.258  5.322 

8.064  5.913 

45 

30 

4.823   3.569 

5.627   4.164 

6.431   4.759 

7.235   5.353 

8.039  5.948 

30 

45 

4.808  3.590 

5.609  4.188 

6.410  4.787 

7.211   5.385 

8.013  5.983 

15 

37    0 

4.792  3.611 

5.590  4.213 

6.389  4.815 

7.188  5.416 

7.986  6.018 

53   0 

15 

4.776  3.632 

5.572  4.237 

6.368   4.842 

7.164   5.448 

7.960  6.053 

45 

30 

4.760  3.653 

5.554  4.261 

6.347   4.870 

7.140  5.479 

7.934  6.088 

30 

45 

4.744  3.673 

5.535   4.286 

6.326   4.898 

7.116  5.510 

7.907   6.122 

15 

38    0 

4.728  3.694 

5.516  4.310 

6.304   4.925 

7.092  5.541 

7.880  6.157 

52   0 

15 

4.712  3.715 

5.497  4.334 

6.283   4.953 

7.068   5.572 

7.853  6.191 

45 

30 

4.696  3.735 

5.478  4.358 

6.261   4.980 

7.043   5.603 

7.826  6.225 

30 

45 

4.679  3.756 

5.459   4.381 

6.239  5.007 

7.019  5.633 

7.799  6.259 

15 

39    0 

4.663   3.776 

5.440  4.405 

6.217   5.035 

6.994   5.664 

7.772  6.293 

51    0 

15 

4.646  3.796 

5.421   4.429 

6.195    5.062 

6.970  5.694 

7.744  6.327 

45 

30 

4.630  3.816 

5.401   4.453 

6.173    5.089 

6.945   5.725 

7.716  6.361 

30 

45 

4.613  3.837 

5.382  4.476 

6.151    5.116 

6.920  5.755 

7.688  6.394 

15 

4O    0 

4.596  3.857 

5.362  4.500 

6.128   5.142 

6.894  5.785 

7.660  6.428 

5O   0 

15 

4.579  3.877 

5.343  4.523 

6.106   5.169 

6.869  5.815 

7.632  6.461 

45 

30 

4.562  3.897 

5.323   4.546 

6.083    5.196 

6.844   5.845 

7.604  6.495 

30 

45 

4.545   3.917 

5.303   4.569 

6.061    5.222 

6.818  5.875 

7.576  6.528 

15 

41    0 

4.528  3.936 

5.283   4.592 

6.038   5.248 

6.792   5.905 

7.547  6.561 

49   0 

15 

f.Sll  3.956 

5.263   4.615 

6.015   5.275 

6.767  5.934 

7.518  6.594 

45 

30 

4.494  3.976 

5.243   4638 

5.992   5.301 

6.741   5.964 

7.490  6.626 

30 

45 

4.476  3.995 

5.222  4.661 

5.968   5.327 

6.715   5.993 

7.461   6:659 

15 

42    0 

4.459  4.015 

5.202  4.684 

5.945   5.353 

6.688  6.022 

7.431   6.691 

48    0 

15 

4.441   4.034 

5.182  4.707 

5.922  5.379 

6.662  6.051 

7.402  6.724 

45 

30 

4.424  4.054 

5.161   4.729 

5.898  5.405 

6.635   6.080 

7.373   6.756 

30 

45 

4.406  4.073 

5.140  4.752' 

5.875   5.430 

6.609  6.109 

7.343  6.788 

15 

43   0 

4.388  4.092 

5.119  4.774 

5.851   5.456 

6.582  6.138 

7.314*6.820 

47   0 

15 

4.370  4.111 

5.099  4.796 

5.827  5.481 

6.555  6.167 

7.284  6.852 

45 

30 

4.352  4.130 

5.078  4.818 

5.803   5.507 

6.528  6.195 

7.254  6.884 

30 

45 

4.334  4.149 

5.057  4.841 

5.779  5.532 

6.501   6.224 

7.224  6.915 

15 

44    0 

4.316  4.168 

5.035  4.863 

5.755   5.557 

6.474  6.252 

7.193  6.947 

46   0 

15 

4.298  4.187 

5.014  4.885 

5.730  5.582 

6.447  6.280 

7.163  6.978 

45 

30 

4.280  4.206 

4.993  4.906 

5.706  5.607 

6.419  6.308 

7.133   7.009 

30 

45 

4.261   4.224 

4.971  4.928 

5,681    5.632 

6.392  6.336 

7.102  7.040 

15 

45    0 

4.243  4.243 

4.950  4.950 

5.657  5.657 

6.364  6.364 

7.071   7.071 

45   0 

0       f 

Dep.     Lat. 

Dep.      Lat, 

Dep,     Lat, 

Dep.     Lat. 

Dep,     Lat, 

0       f 

Bearing, 

Distance  6. 

Distance  7. 

Distance  8. 

Distance  9. 

Distance  1O. 

Bearing, 

45°-60c 


A  TABLE   OF  THE   ANGLES 

Which  every  Point  and  Quarter  Point  of  the  Compass  makes  with  the  Meridian. 


North. 

Points. 

0-1/4 

o-% 

Q-% 
1 

0       f      If 

2  48  45 
5  37  30 
8  26  15 
11  15    0 

Points. 

S3 

»-4 

South.. 

N.  by  E. 

N.  by  W. 

S.  by  E. 

S.  by  W. 

N.N.E. 

N.N.W. 

l—  Vk 

2 

14    3  45 
16  52  30 
19  41  15 
22  30    0 

2 

S.S.E. 

S.S.W. 

N.E.  by  'N. 

N.W.  by  N. 

't\ 
rl 

25  18  45 
28    7  30 
30  56  15 
33  45    0 

l-l 

H 

S.E.  by  S. 

S.W.  by  S. 

N.E. 

N.W. 

1:8 
rl 

36  33  45 
39  22  30 
42  11  15 
45    0    0 

!•:'$ 

J-4 

S.E. 

S.W. 

N.E.  by  E 

N.W.  by  W. 

\--\ 

£* 

47  48  46 
50  37  30 
53  26  15 
56  15    0 

5      * 

S.E.  by  E. 

S.W.  by  W. 

E.N.E. 

W.N.W. 

!:& 
f% 

59    3  45 
61  52  80 
6441  15 
67  30    0 

6 

E.S.E. 

W.S.W. 

E.  by  N. 

W.  by  N. 

6-J4 

«-4J 

7 

70  18  45 
73    7  30 
75  56  15 
78  45    0 

6-4 

ft  —  3/ 

7 

E.  by  S. 

W.  by  S. 

East. 

West. 

7-V4 
8 

81  33  46 
84  22  30 
87  11  15 
90    0    0 

8~   * 

East. 

West. 

THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN     INITIAL    FINE     OF     25     CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  Sl.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


OCT    22  1932 


21  1938 


2    1947 
Y  19    J348 


'     <•*:« 

"3l«>5&ruv 
B 


